Yes it is part of the Jewish tradition- a fact not generally known by Christians!
One of the texts the mystics like to cite as a scriptural allusion to the principle of reincarnation is the following verse in the Book of Job:
Behold, all these things does God do -- twice, even three times with a man -- to bring his soul back from the pit that he may be enlightened with the light of the living. (Job 33:29)
Here is an article on the council held between the 5th May until 2nd June, 553, that changed the view of the Church on the subject of reincarnation:
So you see, reincarnation was part of Christian doctrine (as it was in Jewish doctrine) for the first five and more centuries.
This council produced fourteen new anathemas against the authors of the Three Chapters and other Christian theologians. The eleventh anathema included Origen's name in a list of heretics.
The first anathema reads: "If anyone asserts the fabulous preexistence of souls, and shall assert the monstrous restoration which follows from it: let him be anathema." ("Restoration" means the return of the soul to union with God. Origenists believed that this took place through a path of reincarnation.) It would seem that the death blow had been struck against Origenism and reincarnation in Christianity.
After the council, the Origenist monks were expelled from their Palestinian monastery, some bishops were deposed and once again Origen's writings were destroyed. The anti-Origenist monks had won. The emperor had come down firmly on their side.
In theory, it would seem that the missing papal approval of the anathemas leaves a doctrinal loophole for the belief in reincarnation among all Christians today. But since the Church accepted the anathemas in practice, the result of the council was to end belief in reincarnation in orthodox Christianity.