In addition to the above I think it is important (perhaps because this is my area of study, and I have a bias for it) to add what I see as an emerging way to judge the historicity of our sources. It has been recognized for over a century that the gospels are based (directly or indirectly) on oral traditions. In other words, in the early christian communities, independent sayings, teachings, parables, and stories of jesus were passed around. Thanks mainly to Bultmann, even while his form-critical approach has either largely died out or been very much modified, the effects of his model of orality linger on. A great many scholars simply take for granted that "oral tradition" means freely, uncontrolled, and informal passing along of information about Jesus.
However, an increasing number of scholars have not only challenged Bultmann's model of orality, but the consequences it had.
So with respect to your question, a "newer" way to assess what is more likely to have come from Jesus is to attempt to construct as realistically as possible the model of oral transmission in the Jesus sect (i.e. how information about Jesus was passed along by early christians). This has been done first by looking at other oral cultures, both in general and in particular cultures similar to the one in which Jesus lived, and examining the model of orality in those cultures. Once these models are examined, the christian texts (both in and out of the NT) are examined to see if a similar model can be supported for the Jesus sect.
For example, Kenneth Bailey is not only a scholar, but has also worked extensively as a missionary in remote parts of the middle east, where the culture has remained remarkably unchanged for many years, and is almost completely illiterate. He noticed that different types of information were passed along differently. For example, rumors spread quickly and without any sort of control or reliability (a phenomenon which cross-cultural anthropological research has discovered to be fairly universal). Important information, however, was retained in a particular way. For example, when Bailey would give a sermon, upon completion one of the elders in the community would stand up and say something to the effect of "did you just hear what he said? he said..." and the elder would procede to "package" Bailey's serman in a form which could be easily retained orally. These communities also had their own methods for retaining "Historical" stories about the community. During communal events, a person might ask a question or somehow trigger the recitation of a past event. Not just anyone, however, could tell the story. Certain people were considered "authorities" and were the only ones "allowed" to repeat the story. Additionally, upon recitation, mistakes were corrected by the community. Bailey called this process of transmission "informal and controlled" and theorized that it might be similar to the process of transmission in the Jesus sect.
That is just one, and rather unidimensional, model. I would say most research into oral models of the Jesus sect look at all sorts of models, before seeing if and to what extent they might apply to the Jesus sect.
In any event, by establishing a model of transmission for the Jesus sect, it becomes easier to decide what is a priori more likely to be reliably represented in the gospels. For example, my studies (built so far by looking at anthropological research, rabbinic circles, graeco-roman teacher/disciple communities, and transmission of Islamic ahadith) lead me to believe the following:
1. Jesus' teachings (what he spoke) are by their very nature more likely to be accurately recorded then events which occured during his ministry. Cross-culturally, teachers in oral cultures use repitition and mnemonic devices so students can memorize at least the essentials of their teachings. Jesus (as with all such teachers) almost certainly said the same thing many, many times, probably always or almost always in slightly different ways. Events, on the other hand, only happen once, and are therefore by nature less likely to be accurately recalled.
2. Jesus uses a number of different genres, and each one was probably remembered slightly differently. For example, I short aphorism might be transmitted verbatim. In a longer parable, however, certain details would not be very important. What is important is getting the essentials (sort of like a long joke, where details can change, but if they change too much the punchline doesn't work).
3. If investigations into oral models of teacher/disciple communities and sects (from the Greeks to the early muslims) are any indication, such communities almost NEVER allow anyone to contribute anything to the tradition. Rather, particular people are seen as authoritative, and it is there job to instruct others. In the Jesus sect, there are numerous indications in our texts that such authorities DID exist, and began with the eyewitnesses, particularly those of the twelve who remained with the sect after Jesus death. Luke begins his gospel with the importance of traditions handed down from eyewitnesses. Paul uses particular verbs to refer to "receiving" and "passing on" oral tradition. In addition, Paul spent 15 days "inquiring into" Peter, the head of the Jerusalem community. Papias tells us that he wasn't satisfied with hearing about Jesus from anyone, but first and foremost from eyewitnesses, and next from the disciples of eyewitnesses. John tells us that his source is an unnamed "beloved disciple" indicating again the importance of an eyewitness. Given comparisons with known models of orality and internal evidence, we can plausibly deduce that the Jesus tradition was handled with some care.
4. Number (3) above does not mean that the gospels are highly accurate and perfectly reliable by any stretch of the imagination. For one thing, as has been recognized even before Bultmann, the "story" itself in the gospels (the narrative as a whole) is artificial. The oral tradition was not like a oral gospel, in which a single story was repeated and memorized. Rather, various sayings and short narratives and so forth existed independently of each other. Mark was the first we know of to take many of these pieces and weave them into a narrative. This means that the chronology and context of much of what is said and done in Mark is superficially imposed (which is, by the way, the opposite of myth, where the general story is constant, and the details always change). Additionally, when I contend that the gospels (or at least the synoptics) generally record Jesus' teachings reliably, this does not mean verbatim (although it is possible in some instances). Rather, and again like a joke, the essence of a teaching was important, and details could be changed.
5. Another issue against reliability is authorial redaction. By this I mean the tendency for those "handling" the tradition to alter it to suit their own purposes. Although I think it is likely that this happened to some degree during the oral stage, I don't think it happened to a great extent. As I said, the best model for orality posits a fairly controlled and even at times formal transmission of the tradition. Also, when you are trying to teach someone "Jesus said such and such" there isn't as much need to tailor the saying. This is why, for example, despite the fact that we KNOW circumcision was a very big issue in the early church, NO saying concerning cirmcumcision are placed on Jesus' lips. Beginning with Mark, however, the disperate pieces of tradition were weaved into a narrative (very badly, in Mark's case; his style and greek are really pretty bad). Once this happened, what was a bunch of sayings and stories has to be fitted into one story. Two things happen as a result: one, as already mentioned, is fitting sayings into contexts by having Jesus say them to a particular person in a particular place. The other is the great ability of the author to present Jesus' teachings to make his/her own point. The gospel authors were telling a complete and integrated version of what was really lots of pieces of tradition. In doing so they could put it together to put a particular "spin" on it. They could place a particular saying in a context which would add meaning. They could more easily alter a saying to fit into a context, theme, or theology. They could leave things out that didn't fit their purposes well enough. Or the could add caveats. It is often difficult, then, to sort out where the evangalist ends and Jesus begins, even allowing that the teachings are largely Jesus' own.
6. Very early on in the Jesus sect, and to an increasing degree, the desire to no more about Jesu grew. The "eyewitnesses" after all, were only witnesses to the mission at the end of Jesus' life. And even in the mission there were blank spots (for example, it was easy enough for followers to find out that Jesus was taken to Pilate, but next to impossible to find out what was said). As time went on, we can see an increasing desire to fill in the blank spots. For example, Mark has no birth narrative. But both Matthew and Luke do. And this tendency grows, as with an infancy gospel in which Jesus' childhood is covered.
