Update (30 September 2021): "He was the son of a virgin, from Mary, a Levite. For Mary was from the blood of Aaron." "In referring to Elizabeth as Mary’s kinswoman, the angel refers to Mary relationship to the tribe of Aaron, attributing to her a priestly lineage." (Gregory Nazianzen, Cappadocian Father and Trinitarian Philosopher in I.1.18 (PG 37.480-487) On the genealogy of Christ of “Ordinamento e polimetria,” 179) "Jesus belongs to the royal house of David because of what had happened in Bethlehem: in the city of David, Mary, the Levite, accompanied by Joseph from the house of David, gave birth to the sovereign of the whole universe." (Gregory Nazianzen in Poemata Dogmatica 18: 35-59; PG 37: 483 A-485 A., Ctun non Josepho, sed sit de Virgine natus (Christus ). Ex Maria porro Levi des. Namque ab Arone qui Levi de stirpe fuit, genus ipsa trahebat" (Gregor., Cann.38 de Christi genealogia) Here we see the twisted reasoning that obviously came in problematic afterwards, when the narrative shifted that supposedly Maryˋs genealogy is in Luke 3 - a distortion that started in the writing Doctrina Jacobi chapters 41,42 in 634 a.D. with jews from Tiberia). The problem being that SUPPOSEDLY tribal descend is defined by biological descend - see below - Hebrews 7:14 says: It is evident that our Lord arose from Judah - so Gregoryˋs reasoning that Mary was of mixed blood from Aaron and Judah would bite itself in the tail - the narrative had to shift. Whatever the case may be, Jesus clearly belonged to the tribe of Judah because of Josephˋs davidic descend which Jesus inherited by the marriage of Mary and Joseph)
Mary of the tribe of Levi
We read the following in the book of Hebrews:
"Therefore, if perfection were through the Levitical priesthood for under it the people received the law, what further need was there that another priest should rise according to the order of Melchizedek, and not be called according to the order of Aaron? For the priesthood being changed, of necessity there is also a change of the law. For He of whom these things are spoken belongs to another tribe, from which no man has officiated at the altar. For it is evident that our Lord arose from Judah, of which tribe Moses spoke nothing concerning priesthood. And it is yet far more evident if, in the likeness of Melchizedek, there arises another priest who has come, not according to the law of a fleshly commandment, but according to the power of an endless life."
A frequent problem to be observed in "church circles" (excuse the term, I could not resist) is the doctrinal lens that many people unknowingly read their bibles through. It is interesting to observe that lots of Free Evangelicals flat out mock Roman Catholics for believing such things like "the ascension of Mary" or "the perpetual virginity of Mary" (while there are good reasons to believe the latter, more about that at some other point) without biblical research, but at the same time proclaim dogmas as unshakable truth that they themselves never checked thoroughly and obviously neither were aware of being roman catholic to the core. One of such teachings is that Mary, the mother of Jesus, is of the tribe of Judah and of the house of David. Before I jump into a biblical analysis, let me go ahead and give you a few quotes from well known authors that are regarded as high level theologians in the world of theological research (not necessarily my world, but you cannot escape dealing with it). As such, there are quotes from the roman catholic Raymond E. Brown, lexicographer and theologian Walter Bauer and protestant theologian Theodor Zahn.
"Gentile writers often did not understand how Jesus could be truly a Davidid through Joseph who did not beget him and so were forced to think of Mary as a Davidid." 
"If Mary was of the house of David, why would it be necessary to tell readers about Joseph's lineage? Later church writers did attribute davidic descend to Mary (not necessarily through the lucan genealogy); but that often stemmed from an inability to understand that in a jewish mindset, through Joseph's acknowledgement, Jesus could be legally, even if not biologically, Joseph's son and thus share davidic descend." 
"Yet we already encounter voices in antiquity that declare Mary to be a Levite." 
"In order to name Jesus a David-son, one believed that Mary, through whom he [supposedly] soley is connected to human nature, must be of the same house. Zahn (Forschungen I p. 265) chooses strong words for the crudeness of this 'fiction from gentile christians'." 
"Jesus was erroneously thought to be the son of David...he was this according to the law...The legal paternity of Jesus has importance...because only if Jesus is the son of a Davidian, he himself is a David-son." 
"I do not deem it necessary to prove afresh that Mt. 1:1-16, Lk. 1:27, 2:4, 3:23-31 merely state factually the davidic ancestry of Joseph, as well as that the status in the eyes of his people as the messiah and son of David are based on his never challenged legal sonship of Joseph, as whose son he was regarded by friend and enemies alike (Jn. 1:45). The fable of the davidic ancestry of Mary appears first in the Protogospel of James, then in Justin Dialogues with Trypho 43, 45, 100, 120...His friend Tatian then forced it alongside Lk. 2:4 into the Diatessaron. From the 3rd century on, this was almost the sole viewpoint. But as little as it is found in the New Testament, there is not found a trace in the jewish christian literature. (Gospel to the Hebrews, Ebionites, Clement, Hegesipp)" 
"He who...claimed Mary to be of davidic ancestry...had to find artificial skillful means for proving it." 
"No in depth analysis of this and similar claims is necessary to see that this is a very ancient distorted idea [the idea that Mary and Joseph had levitical and davidic ancestry as a requirement for the right of Jesus to the throne and priesthood], based on an assumption of the davidic ancestry of Mary, which was further connected to the original thought that Mary was a Levite." 
I could bombard you with more quotes, but the mere reason for quoting these men is to show the reader that first of all, the idea that Mary was a Levite is more ancient than the idea that she is of Judah, and also that it was due to the distorted thinking of gentile christians that Mary was somehow "required" to be of the house of David. I would go one step further and claim that satan himself wanted this idea to become the gold standard because then he could throw the hypostatic union in the mix and essentially deny that the Word of God was literally killed at the cross, something he himself and his fellows did not fully comprehend when it was executed (see 1. Corinthians 2:7-8) that however disarmed him and his angels (see Colossians 2:13-15).
Let us try to keep this study as short as possible and start by stating the fact that a davidic ancestry of Mary is nowhere to be found in the bible. There are some voices that want the lucan genealogy to be of Mary (though it plainly says that it is of Joseph), but even here we are confronted with the fact that there are levite names in the genealogy. (see also Raymond E. Brown's book page 588) The lucan genealogy further is not a flesh and blood tracking record as it terminates with "...of Seth, of Adam, of God". It clearly is a legal genealogy of Joseph since Adam is not a flesh and blood offspring of God. Further, it was a known fact that the discrepancies between Matthew's (flesh and blood) and Luke's (legal) genealogies are diverting among other reasons due to levirate marriages. This was the view prior to 634 a.D., when a writing called "Jacob newly baptized" (Doctrina Jacobi) popped up. In this polemic writing, unbelieving Jews from Tiberia recite their (!) legend which states that Luke gives Mary's genealogy. It is mind boggling, to what length theologians go in order to validate such a claim. As such, the note that Jesus "was supposed to be the son of Joseph" (Luke 3:23) is taken as some legal term that really means that you now are about to read the genealogy of Joseph's step father and thus the genealogy of Mary. Even Strong's lexicon in G3543 "nomizo" misleads the reader by giving a supposed prime meaning of the term to some "legal supposing" by citing one instance (!) in Acts 16:13, where the thinking person quite easily can understand that laws actually become customs by what people suppose. The secondary meaning (as Strong's wants to make us believe - it is actually the prime and sole meaning) of the term has the full support of 13 (!) examples in this entry. And what are they? That people actually think stuff to be a certain way. No way that this could be the meaning of Luke 3:23, could it?
"Is this not the carpenter’s son? Is not His mother called Mary? And His brothers James, Joses, Simon, and Judas?"
"However, we know where this Man is from; but when the Christ comes, no one knows where He is from."
"And they said: Is not this Jesus, the son of Joseph, whose father and mother we know? How is it then that He says, ‘I have come down from heaven’?"
I apologize upfront, but if you like to escape biblical reasoning and open honest analysis, then the kingdom of God is not the place you want to hang out at. There is nothing covered that will remain hidden, so why should we shy away from an honest contemplation of this topic? You have to be honest please and admit that satan is at work, crafting, hiding contrary evidence, creating hellfire-bound church dogmas and excluding those from the fellowship who dare to raise their opinions based on findings, more so, stating that in fact it is a LIE that Luke 3 lists the genealogy of Mary. It simply is untrue. It took the hardcore Mary worshipper John Damascene to further this jewish legend. Later, it was John Wesley that made it go mainstream in the church world of protestant denominations to the point that the person today claiming Mary to be a Levite is looked at like some sectarian alien that landed with a spaceship from another planet. However, the bible clearly tells us that Mary in fact was a Levite. Let us gently approach some good solid biblical reasoning as to why that is.
"There was in the days of Herod, the king of Judea, a certain priest named Zacharias, of the division of Abijah. His wife was of the daughters of Aaron, and her name was Elizabeth."
As we see, the mother of John the Baptist was "of the daughters of Aaron". Whilst analyzing counter arguments, the flat out dumbest one was that Elizabeth was in fact also of the house of David and somehow became a daughter of Aaron by her marriage to Zacharias. Really? Luke must be a treasure box of hidden gnostic insight. While a wife certainly becomes a relative belonging to the husbands ' tribe by marriage, it is beyond reasoning to claim that this makes her a daughter of the racial progenitor of the tribe. Also, it says that "his (Zachariah's) wife (Elizabeth) was of the daughters of Aaron". Why state it in this manner, if she became a daughter of Aaron by her marriage to Zachariah? There is further a statement from the relatives of Elizabeth in Luke 1:59-61. They wanted her to name John "Zachariah", reasoning that none of her relatives were named John. There is one high priest mentioned in Acts 4:6 that is named John. Either he was born after Elizabeth birthed John, or this high priest John was of a different division. We know that Zachariah was of the division of Abijah. Bearing in mind that the relatives of Elizabeth wanted John to be called Zachariah based on Elizabeth's family names, we have more proof that she in fact belonged to the tribe of Levi, not Judah as some would like to have it in order to declare Mary a Davidid. We can safely believe that Elizabeth was BY BIRTH a "daughter of Aaron", scripture very clearly tells us this. You see, churches love keeping stuff volatile whenever it suits their case. Then why craft hellfire-bound dogmas for all who do not eat their doctrines? Looking further down to verse 36, we learn this now:
"Now indeed, Elizabeth your relative has also conceived a son in her old age; and this is now the sixth month for her who was called barren."
The story further recounts how Mary goes ahead and visits this relative of her named Elizabeth and spends a full 3 months with this family of the house of Aaron. This should already be somewhat convincing that there is no "judaean ancestry from the house of David" with Mary. Would it not be outright degrading to her messianic davidic family to spend a full 3 months with some cousin of hers, rather than sharing the good news of the advent of the long expected messiah with the davidic family of hers? She did not, and further went home after those 3 months. The angel Gabriel told her prior to this that Jesus will be given "the throne of HIS father, David" (v. 32). Why not "the throne of YOUR father David" when Mary is addressed as a Davidid of the house of Judah?
The apparent problem for the church (I am not talking abut the body of Jesus Christ, I believe "church" is a pagan term - more on that in a later article) now is evident: Mary was related to Elizabeth. Origen already attempted to shake this off stating that benjaminite Paul was a relative to all Israelites in Romans 9:3. As if Luke 1:5.36 could mean that Elizabeth was of the tribe of Dan and Mary of the tribe of Zebulon. But why would God give us the levite ancestry of Elizabeth and call Mary her relative if there was no importance in this?
"Every word of God is pure; He is a shield to those who put their trust in Him."
God has given us the information to defend the truth and to break to pieces all of satan's lies. His Word is actually a hammer. The misguided kabbalistic jewish mystic will seek to inform us that kinship is derived from the mother. Let me quote from Wikipedia if I may.
"Orthodox Jews, who believe that matrilineality and matriarchy within Judaism are related to the metaphysical concept of the Jewish soul, maintain that matrilineal descent is an oral law from at least the time of the covenant at Sinai (c. 1310 BCE). Conservative Jewish Theologian Rabbi Louis Jacobs suggests that the marriage practices of the Jewish community were re-stated as a law of matrilineal descent in the early Tannaitic Period (c. 10-70 CE)" 
So we are dealing with an oral law that for some reason popped up right in or after the days of Jesus Christ. Strange. While I am not placing any trust in the jewish Talmud, I cannot get around citing a quote from it for the reader.
"It is the father’s family that is called one’s family, while one’s mother’s family is not called one’s family. Proof for this is found in another verse, as it is written: “By their families, by their fathers’ houses” (Numbers 1:2)."
(Bava Batra 109b)
Let us quote the full bible verse here:
"Take a census of all the congregation of the children of Israel, by their families, by their fathers’ houses, according to the number of names, every male individually."
Now let us analyze first the Wording of Luke 1:36 from the interlinear bible.
As we see, the word for "relative" is Strong's number G4773 "suggenes", meaning "of the stock/generation". The New Testament hebrew Delitzsch text has H7126 "Near" which implies close kinship of blood relation (see the book of Ruth 2:20 that has H7138 which derives from H7126).
Somehow, the Elizabeth of advanced age was related to the apparently younger Mary. This word implies blood relation, whereas care is to be taken since marriage for a female is a means of belonging to the stock of another tribe. Since we are contemplating a clearly aaronite born woman though (Elizabeth), being a "suggenes" of the unmarried Mary, we are left with the only conclusion that Mary must have had a levite father. (Levi was the great grandfather of Aaron) This is due to the fact that kinship is derived from the father, not the mother. Some dance around the cake and claim that maybe Mary's mother was aaronic and her father davidic and that somehow made Mary the cousin of Elizabeth, who then was the sister of some "Anna", a fictitious character that serves as Mary's mother in extra biblical accounts. Let me quote Numbers 1:2 in the Septuagint.
As we see, the "suggenes" go according to the patrimony. This verse already includes the word "father". Suggenos is determined by the father, not the mother. Since Elizabeth was already married to Zachariah, she counted as an Aaronite even if she supposedly was a Dividid of the house of Judah by birth that somehow became a daughter of Aaron by marriage. Of course that assumption is outrageously false. Much rather, the patrimony determines the origin of a person, the generation, the genealogy (G1078 as used in Matthew 1:1 about the Lord Jesus Christ).
Since Paul in Romans 9:3 talks about "Israelites" as his "suggenes", we now understand that by the patrimony of Jacob (who is Israel), the benjaminite Paul is very well counting to that patrimony of the house of Israel. But the bible would certainly not state that Elizabeth was of the daughters of Aaron and then mention her to be stock-related to betrothed Mary who goes ahead and spends 3 months there, if there is no clear statement here that Mary was an Aaronite. I have proven that it is unbiblical to assume her mother to be the connecting element and shown that it borders to intellectual suicide to assume that Elizabeth is of the house of David. The clearest sense of the bible is made if Elizabeth was Mary's aunt on her father's side - the sister of the high priestly father of Mirjam (the hebrew levite name of Mary). It was nothing out of the ordinary when high priest daughters married sons of kings. There are a total of 3 of such instances in the genealogy of Joseph of Matthew 1 alone. Do this study with the Old Testament and Matthew 1 to see that it is true. You might find that 1 Chronicles, a book that used to bore you to death, becomes your new favorite, helping you understand not only bible prophecy but also high priest lineages. I mentioned the levite names in Luke 3 and I should also mention that the high priest Phinehas was the grandson of the marriage of a Levite and the daughter of the tribal leader of Judah. There is a whole arsenal of statements in the Testament of the 12 patriarchs about an expectation of a davidic-aaronite messiah. Not that it would further my case since it is extrabiblical material, but it shows that these ideas are not some modern claim. They are deeply rooted in messianic expectations. The lines of David and Aaron have been mixed throughout the centuries.
This now brings us to a problem for the "marian flesh" proponents (there are many, too many in my opinion). Hebrews 7:16 clearly states that Jesus Christ did not inherit the priesthood by a fleshly inheritance. One shred of aaronite flesh in Mary would count against their theory because in their mindset, whatever maternal flesh Jesus is built of, that determines his tribal affiliation. The priesthood was changed and thus the law was changed. Let us examine a few passages:
"Phinehas the son of Eleazar, the son of Aaron the priest, has turned back My wrath from the children of Israel, because he was zealous with My zeal among them, so that I did not consume the children of Israel in My zeal. Therefore say, ‘Behold, I give to him My covenant of peace; and it shall be to him and his descendants after him a covenant of an everlasting priesthood, because he was zealous for his God, and made atonement for the children of Israel."
"Then Phinehas stood up and intervened, and the plague was stopped. And that was accounted to him for righteousness to all generations forevermore."
"Thus says the Lord: ‘If you can break My covenant with the day and My covenant with the night, so that there will not be day and night in their season, then My covenant may also be broken with David My servant, so that he shall not have a son to reign on his throne, and with the Levites, the priests, My ministers. As the host of heaven cannot be numbered, nor the sand of the sea measured, so will I multiply the descendants of David My servant and the Levites who minister to Me.’ "
"And behind the second veil, the part of the tabernacle which is called the Holiest of All, which had the golden censer and the ark of the covenant overlaid on all sides with gold, in which were the golden pot that had the manna, Aaron’s rod that budded, and the tablets of the covenant."
The argument is firmly established that there are in fact 2 covenants of messianic character in the Old Testament. They are further reflected in the book of Zechariah with Yeshua the high priest and Zerubbabel of the house of David. Clearly, God says that he will break neither of them. So either we will see a bunch of levite priests in the millennium that sacrifice bulls, goats and an annual lamb, or this covenant merged into Jesus Christ alongside his davidic covenant. (There is no hint here that I reject a future millennium by the way)
The promise to Phinehas is twofold. God promised HIM and his descendants an everlasting priesthood and God reiterates this in Jeremiah 33. The missing piece of information is that the book of Hebrews compares the tabernacle of the desert to the body of Jesus Christ.
"But Christ came as High Priest of the good things to come, with (really: 'by means of' - in greek 'δίά') the greater and more perfect tabernacle not made with hands, neither of this creation."
"Therefore, brethren, having boldness to enter the Holiest by the blood of Jesus, by a new and living way which He consecrated for us, through the veil, that is, His flesh..."
Note the clear statement that the body of Jesus Christ is not out of this creation and also not made with hands. The veil constituted his flesh. We learned that Aaron's rod was in the ark of the covenant, something that only Hebrews 9 mentions in the bible. It is not far fetched to see this as the fulfillment to merge the aaronic priesthood into the messianic kingship of Jesus Christ. After all, he is our high priest and we ourselves are a royal priesthood. It is the christians that have inherited these promises of God. I am not saying that God is not dealing with the inhabitants of Israel at some future time, this is not the topic, I do very well believe that God has wonderful plans for this country and one third of those who will happen to be living there in the time prior to the return of Jesus Christ. I however am firmly convinced that the law of aaronic priesthood was changed and this covenant merged into Jesus. There is a good possibility that this is because of the mentioned merging of tribal inheritance through the mother in case of missing siblings (see Numbers 36). If we assume that Mary had no brothers and her father was the only high priest left of that tribe, himself without brother, then Mary would have merged the aaronic priesthood into Jesus by default through her marriage to Joseph. Genesis 49:10 and Daniel 9:26 in the Septuagint provide strong language that indicate the termination of the priesthood with the advent of the messiah (see "On the priesthood of Jesus" and footnotes in Burke & Landau "New Testament Apocrypha, more non-canonical writings")
There is another point that has less weight which is that Mary as a Levite would have a claim to carry the tabernacle. 1.Chronicles 15:2 and Numbers 1:50 reserve this right to Levites alone. As the greater and more perfect tabernacle is the body of Jesus Christ, she qualified to carry it.
The prophecy to the "alma" (young woman) of Isaiah 7:14 "Thou shalt conceive in thy womb and bear a Son" which is applied further to Mary in Matthew 1:23 is to a high degree likely to be applied to Abijah, the high priest daughter and future mother of King Hezekiah. So this prophecy was uttered pertaining to a Levite in immediate fulfillment as well as in fulfillment of a Levite to come - the virgin Mary. Raymond E. Brown mentions in his book on page 116 that Miryam of Levi (sister of Moses) received a crown of royalty as per Targum Neophiti in Numbers 1:15, and that she further was an ancestor to King David. I would not be dogmatic about these points, but they round off the whole study to a nice picture, so I mention them.
Mary was a Levite. This is a teaching found in the bible and it is highly consistent. It utterly destroys the view that the Lord Jesus Christ had marian flesh. This is why the church cannot accept it to be true - it sinks the ship of church father claims down to the council of Chalcedon and exposes, once again, their dogmas to be pagan satanic ideas.
Update (30 September 2021):
(Gregory Nazianzen, Cappadocian Father and Trinitarian Philosopher in I.1.18 (PG 37.480-487) On the genealogy of Christ of “Ordinamento e polimetria,” 179)
(Gregory Nazianzen in Poemata Dogmatica 18: 35-59; PG 37: 483 A-485 A., Ctun non Josepho, sed sit de Virgine natus (Christus ). Ex Maria porro Levi des. Namque ab Arone qui Levi de stirpe fuit, genus ipsa trahebat" (Gregor., Cann.38 de Christi genealogia)
 Raymond E. Brown "The birth of the messiah" p. 288
 Raymond E. Brown "The birth of the messiah" p. 589
 Walter Bauer "Das Leben Jesu im Zeitalter der neutestamentlichen Apokryphen" p. 10 (free translation)
 Walter Bauer "Das Leben Jesu im Zeitalter der neutestamentlichen Apokryphen" p. 15 the Word "supposedly" was interjected in order to express the german Konjunktiv I (free translation)
 Theodor Zahn "Forschungen zur Geschichte des neutestamentlichen Kanons und der altkirchlichen Literatur Vol. 6" p. 328 (free translation)
 Theodor Zahn "Forschungen zur Geschichte des neutestamentlichen Kanons und der altkirchlichen Literatur Vol. 6" p. 328 Footnote (free translation)
 Theodor Zahn "Forschungen zur Geschichte des neutestamentlichen Kanons und der altkirchlichen Literatur Vol. 6" p. 329 (free translation)
 Theodor Zahn "Forschungen zur Geschichte des neutestamentlichen Kanons und der altkirchlichen Literatur Vol. 6" p. 329 Footnote (free translation)