Although every single academic scholar teaching in this area, who I posed the question to, has written me back that there is no surviving evidence of reincarnation being taught by religious groups during the time of Jesus, it is clear to me that not only was it taught, but it was actually taken for granted by Jesus, the disciples and other spiritually knowledgeable people of the time.(1) Articles like the one this comment is linked to demonstrate the variety of teachers and teachings which embraced an understanding of reincarnation.
However, I should point out that what survives of these teachings is not entirely accurate as regards reincarnation itself. I suspect that the more accurate teachings have been distorted or lost altogether, and what survives is a kind of hybrid.(2) So I create this link to show that reincarnation is hardly a new teaching historically in Christianity; but not to present the teachings themselves as being entirely correct in all respects. More accurate teachings can be found in my list of Recommended Books.
1) Note the following passage, #109, from the Gospel of Thomas from the Nag Hammadi documents:
Jesus said, "The kingdom is like a man who had a hidden treasure in his field without knowing it. And after he died, he left it to his son. The son did not know (about the treasure). He inherited the field and sold it. And the one who bought it went plowing and found the treasure. He began to lend money at interest to whomever he wished." (Thomas O. Lambdin translation) reference
Unless I miss my guess, this is a clear reference to reincarnation. The hidden treasure is the true Self or Atman, which is essentially One with God. "Without knowing it" means that the man is not conscious of this Self. Inheriting and selling the field means, reincarnating, reinheriting the body and the physical world of experience. We have three incarnations represented here for purposes of illustration, but of course the actual process takes many more lifetimes. "Plowing" means directing consciousness inward, seeking within oneself. Finding the treasure means finding the Self, becoming consciously aware of the Self within. "Lending money at interest" means teaching and spiritual work after attaining Realization, i.e., giving out of one's spiritual wisdom and power to raise up others, which then yields spiritual "interest."
2) I am forced to say the same thing about the Jewish teachings on reincarnation that I have seen, including the links I've presented above this one, that what has come down to us through that tradition appears to be an admixture. For example, I think it's a fundamental error to suppose that the soul can return to "nothingness" and that consciousness could decay as the body decays, due to over-identification with the body. Over-identification with the body can certainly cause the person who has died to continue to cling to it and might thus cause them to be earth-bound for a period of time. It's a small point but a highly significant one, inasmuch as if any soul can "disappear" like this it throws the entire moral order and purpose of Creation into question. This is just one example. Another would be the teaching that Jews reincarnate only (or primarily) as Jews. It may be true for a few incarnations in a row, due to strong identification with the culture, but in the longer scheme of things it would fly in the face of one of the primary purposes behind reincarnation, to gain well-rounded experience and eventually identify with something more fundamental in human nature than cultural or religious affiliations.