If Mark is a primary source behind Matthew and Luke, then we still have to explain where the material common to Matthew and Luke that is not in Mark comes from. For some, it is just easier to go with Matthew as the first gospel. Then Mark and Luke can simply be an abbreviation and rearrangement of Matthew respectively. But if the last post about Markan priority holds true, then we will need an alternative explanation.
The "common oral tradition" explanation does not seem, at least to me in my limited exploration, to be able to account for the amount of this common material (remembering that Jesus did a lot more than what is recorded) and the extent of its verbal similarity (although my sense is that this material is less close verbally than the Markan material). In any case, by far the majority position is that some written source explains this material.
There are two main alternatives. The one sees Luke drawing directly on Matthew and Mark. The other sees both Matthew and Luke drawing on some common source or sources, primarily a collection of Jesus' sayings. Gospel experts are less united on this one. Probably a majority still go with a common source, a collection of Jesus' sayings. It's usually called "Q," which is short for Quelle, "source."
Renewed studies of oral tradition do complicate these hypotheses. Despite any written sources that may have been in play, Matthew or Luke's memory of sayings and events was probably also a factor in how things ended up like they did. Also, even if Luke had a written source of Jesus' sayings in front of him, he might also have a copy of Matthew in front of him or have heard Matthew at some time. If we knew the complete story, I imagine it would be pretty complicated.
So here is my current sense of things as someone who has done some study but who is not an authority on them per se. Yes, there are some places where Matthew and Luke agree in wording against Mark, sometimes used to argue that Luke knew Matthew even though he primarily followed Mark. In the opinion of most, these instances are not significant enough to dislodge Markan priority. However, some believe they are significant enough to make it unnecessary to hypothesize some sayings source (e.g., Mark Goodacre). Rather, they suggest that Luke had Matthew in addition to Mark and that Luke's sayings material comes from Matthew rather than some Q.
On the one hand, I am very open to the possibility that Luke knew Matthew. However, in the end, I am still siding with the majority that the evidence ultimately suggests Luke also had a sayings source that Matthew also used. I offer a few reasons why I think this majority position--again, a position that has stuck around for well over 100 years--seems more likely.
First, if we look at Matthew and Luke's common material, sometimes Matthew's version seems more original and sometimes Luke's version seems more original. This speaks to a common source rather than to one of them using the other. On the one hand, I don't know any voice of any significance that argues that Matthew used Luke. It can be argued, of course, but look at the way that Luke arguably edits this saying that is also in Matthew:
Do not be afraid of those who kill the body, but are not able to kill the soul, but rather be afraid of the one able to destroy both soul and body in Gehenna.
Do not be afraid of those who kill the body and after these things are not having something further to do. I will show you whom you should fear. Be afraid of the one who, after killing, has the authority to cast into Gehenna.
Here, it seems to me, Matthew preserves the more original wording and Luke has expanded on the saying. The real question is thus whether Luke at some point has a more original version of a saying and Matthew seems more to paraphrase it. Consider the following possibility:
Blessed [are] the poor in spirit, because theirs is the kingdom of heaven.
Blessed [are] the poor, because yours is the kingdom of God.
It is hard to argue for which might be original. The poor are a major theme of Luke, so one might argue that it might despiritualize Matthew's version. On the other hand "kingdom of heaven" is a clear redactional tendency of Matthew, something he might modify from Mark. In the end, I think it slightly more likely that Matthew would spiritualize the saying than that Luke would despiritualize it.
But the strongest reason why I think Luke does not get his common sayings material from Matthew is the fact that I can't think of any reason he would scatter Matthew's teaching. Take Luke 16:14-18. This material appears in three different places in Matthew. Matthew's presentation of it is magisterial and great pedagogically, appearing in coherent sermon contexts. In Luke, though, even the NIV puts as the heading for these verses, "Additional Teachings." They just seem to come out of nowhere and to be grouped somewhat randomly.
In short, it makes sense to me that Matthew might take a sayings source and organize some of its material together. It makes no sense to me that Luke would look at Matthew and partition its sayings material. For this reason, a majority still leans toward the over a century old suggestion that both Matthew and Luke drew on a common source, consisting mostly of Jesus' sayings, for this common material, called "Q" for short. Luke's order for this material is often taken to be closer to how it might have actually appeared in this source than Matthew's order.
Now if you follow this line of thinking, it leads to some thoughts that might feel a little uncomfortable. I am not going to die for any of them but you can see that they are not simply some whacko liberal clap trap. They are the result of experts doing detailed analyses and following the evidence to what seems to be its most logical conclusions. These are not beyond question for certain, but they're also certainly not some attempt to undermine the Bible.
For example, the Sermon on the Mount would turn out to be in part a collection of Jesus material, with Luke's Sermon on the Plain as its core. Matthew as we now have it would probably not turn out to be in the same form as the gospel the second century Papias mentions--a collection of Jesus' sayings in Aramaic. Although it is not a popular suggestion these days, I myself like to think that what we call "Q" was actually an expanded Greek version of an Aramaic collection of Jesus' sayings that the disciple Matthew made. I can't prove it, of course.
The Gospel of Matthew would thus be named after one of its primary sources. We do have examples of these sorts of sayings collections. The Gospel of Thomas is one, as are 4QTestimonia and 4QFlorilegium from Qumran (although they're a little different from what we are picturing here).
In any case, this is some of the stuff that gospel experts assume and some of the stuff they debate...