The most prominent recent attempt to argue for Lukan Priority is the work of Robert Lindsey. His work, A Hebrew Translation of the Gospel of Mark, is available here, and has been taught & expounded by the organization Jerusalem Perspective.
Lindsey did not intend to become a Synoptic Problem scholar, but stumbled onto a discovery that led him to challenge the consensus of his time. In creating a Hebrew translation of the Gospel of Mark for his congregation, Lindsey (who at the time believed in Markan Priority--like most of us, it's what he was taught) discovered the following patterns:
- Portions of Mark translate well into Hebrew, as if the
original syntax of the material was Hebrew, not Greek. However, this
is very inconsistent, and much of Mark lacks this syntactical
- Luke translates beautifully into Hebrew--better than do any of the other Gospels (Tresmontant came to the same conclusion in The Hebrew Christ), suggesting this is material that was originally composed in Hebrew, and has been translated carefully into a Greek that preserves the Hebrew sentence structure
- Portions of Matthew translate into Hebrew about as well as Luke does, but the portions of Matthew that overlap with Mark follow essentially the same pattern Mark does (see #1)
The Historic Present
Mark regularly employs the "historic present"; he'll talk about the past and say things like "they were saying", whereas Matthew & Luke more often use the customary "they said". This is a noteworthy stylistic feature of Mark (often lost in translation if not read in Greek) that occasionally shows up in the other Gospels.
Lindsey concluded that Luke's editorial practice makes no sense if Luke is drawing material from Mark, but Mark's preference for the historic present is explicable if Mark is drawing material from Luke. Luke isn't afraid to use the historic present--he does so a number of times--but he almost never does so in places where Mark does. If Luke came after Mark, that would mean Luke methodically edited out Mark's historic present almost every single time, but then introduced it himself in other passages.
Mark, on the other hand, could have added this feature--it was just one of his personal mannerisms--but didn't copy every story in Luke (Mark is much shorter than Luke), so sometimes Mark didn't include Luke's historic present because it's found in a story that Mark didn't use.
Lindsey suggested the existence of several hypothetical documents to get the material from the eyewitnesses to the first (surviving) Gospel. He further suggested that the incredibly Hebraic nature of Luke's Greek is best explained by Luke writing prior to Matthew & Mark--Luke's material is closer to the source and more original--than it is by Luke having consciously edited back into a Hebrew style after reading Greek Mark.
Lindsey's hypothesis puts Mark 2nd, reliant upon Luke and a hypothetical source, and Matthew 3rd, reliant upon Mark and a hypothetical source. In this way the less Hebraic-Greek of Mark influences Matthew, but in places where there was no Mark to follow, Matthew is more faithful to the Hebraic structure of his other source.
Lindsey's hypothesis has the advantage of working with the Semitic nature of the material in a way that most hypotheses on the Synoptic Problem ignore--by conducting all comparisons in Greek. Hebrew scholars like Lindsey, Tresmontant, and Carmignac have adeptly pointed out Hebrew structure in the Gospels that is not well-explained by oral hypotheses or existing source-criticism consensus.
A more thorough review of Lindsey's hypothesis is presented on YouTube here by the Jerusalem Perspective, as taught by his student Halvor Ronning.
My own work on the Synoptic Problem builds upon that of Lindsey, but applies the evidence of the Hebrew language without the need to appeal to hypothetical documents (it also comes out against Lukan Priority).