Not true. Early Jewish sources, such as the Talmud, attest to the fact that the priestly class of Jesus' time wanted Him dead. In fact, the Talmud doesn't mention the Romans at all; they give full credit to the priests for hanging Yesu"(sorry, don't have the exact quotes in front of me, but they're readily available on the Internet and at any library).
P.S. I'm not saying that the Romans weren't the ones who executed Jesus, simply stating the fact that the Talmud seems not to be shy in saying that the High Priests were instrumental in His death.
The Talmud does describe the execution (by stoning, not crucifixion) of a Jewish heretic named Yeshu, but it is not talking about Jesus of Nazareth; the Talmudic story describes an event that took place during the reign of "King Yannai" [Alexander Jannai], a Jewish king of the Maccabean dynasty, who reigned from 105 B.C.E. to 75 B.C.E., long before the time of Jesus of Nazareth.
In Jannai's time, Judea was independent and the Sanhedrin could indeed sentence people to death; in Roman times, only the Roman governor could sentence someone to death. Josephus's Antiquities, the earliest history to mention Jesus outside the New Testament, says that of Jesus that "Pilate sentenced him to be crucified."