Cynicism has traditionally been attributed to Antisthenes, a student of Socrates. I'm not very familiar with Antisthenes, but the main current of Socrates' thought was extreme skepticism about all knowledge; the oracle said that there was no one wiser than Socrates, which Socrates, after some initial confusion, took to mean that he was wiser simply because the others didn't realize that they lacked wisdom, whereas he knew that he lacked it. One of the things he got in trouble for was saying that he didn't know whether or not traditional Greek myths such as the ones in Homer were true. This overwhelmingly negative philosophy doesn't seem very consistent with Jesus' conviction that he knew all the answers and that the ultimate truth he revealed was accessible to ordinary people.
Socrates and Antisthenes also considered virtue to be either its own reward or to be justified because it would automatically lead to happiness (defined as pleasure, the satisfaction of desires, and the fulfilment of one's nature). I assume Jesus would have seen virtue as being either rewarded by God's favor after the end of the world, or as being a side issue that would be neither necessary nor sufficient for God's favor.
Other names associated with the Jesus-as-cynic theory are Gerd Theissen and Leif E. Vaage. A 1996 article in the Atlantic by Charlotte Allen has quite a few details about the idea and also on why a lot of people think it's silly.
All I've read by Crossan is Jesus: A Revolutionary Biography, which is a popularization. It doesn't assert that Jesus was a Cynic, it just makes a loose analogy. Cynics of this period, 4 centuries after Antisthenes, had a specialized costume, including a double cloak, a type of purse(/wallet/knapsack) and a staff, that made them recognizable stock characters of the period, and they were urbanites. Jesus came from a tiny agricultural hamlet and spent his time tramping around rural areas. Cynics scorned authority, whereas Jesus, and the numerous "false messiahs" of the period, said that they wanted to reimpose God's authority (basileia, usually translated as kingdom but literally meaning "rule" or "power").
Jesus was no doubt familiar with the Cynics, as well as with other groups like the Essenes, and he may very well have been influenced by some of their ideas. But there is no historical record of his having been considered by his contemporaries as being associated with these groups. He did not wear the uniform of a cynic, and so would not have been identified as one. Both the NT and historical sources such as Josephus identify him as being much more recognizable to people of the period as a different and equally well defined type, which was the kind of rabble-rousing preacher who claimed to be the messiah and whom the Romans labeled a bandit (λῃστής, with Robin Hood overtones to their followers and political overtones to everyone).
However, the type of wisdom sayings found in the Sermon on the Mount are the kinds of things you would expect a Cynic to say. Jesus was also similar to the Cynics, and unlike the other messianic preachers of the period, in that his mission consisted in large part of the lifestyle that he demonstrated, which was an assault on traditional family, social, gender, and ethnocentric values. Mark 6:8 has Jesus telling the apostles "that they should take nothing for their journey, except a staff only: no bread, no wallet, no money in their purse, but to wear sandals, and not put on two tunics." This is a command not to wear the stereotyped uniform of a cynic.