In defense of peacefulpete, there really is nothing wrong with the answer pete provided. It is not the main reason, but it isn't incorrect. (Consider it a "besides," as some call it.)
Think of it this way, as I think we can all relate to what it used to be when we were all self-centered in the world of Jehovah's Witnesses:
Now that we are out of the Watchtower, we realize that nothing we did, no publication we released or any event we engaged in was really noticed by anybody. For example: remember those super-big Yankee Stadium conventions that Jehovah's Witnesses held once and were supposedly history-making that "the entire world took note of"? Nobody really knows about about those. Nobody cares. Conventions where? Who? What? They meant nothing to the world.
Now let's apply this to the Trinity and why we Jews don't believe in the concept. It has to do with the historical Jesus of Nazareth. According to secular history and that of my people the Jews, Jesus, son of Joseph, the rabbi from Nazareth did indeed exist. The only difference is that for us Jews, Jesus was like a Watchtower religion event occuring in our Jewish world--we didn't notice much of anything while he was happening.
In fact, we Jews had a different messianic concept than what Christians eventually developed due to what happened when Jesus redefined his followers expectation when he died on the Cross (and redefine those expectation, Jesus did--but that is a different story I won't go into here and now).
The Jewish messianic concept developed when the Hasmonean dynasty failed due to falling into the hands of the Herods, and the Herods betraying the Hasmoneans by selling them out to the Romans who overtook our land in exchange of putting the Herods onto the throne. The Jews, always looking for a divine reason why national tragedy occurs blamed this one on the fact that the Hasomean kings were all Levites instead of Judeans. The Mosaic Law and other texts state that rulership should be in the hands of the House of David, and thus a "reason" for God's "punishment" was seen in the Roman oppression.
The only way out was that God would send a deliverer, Jews thought, a Davidic king anointed to as prince to save the people and remove the yoke of oppression. Thus the hope of a restored kingdom, like the one the Hasmoneans had tried to build, but with a son of David as ruler. (This was newly discovered and retrofitted from re-reading the Biblical texts to fit into the new Roman oppression. There are no Hebrew Scriptures that mention a figure called "the Messiah.")
But this prince would be more like Judah Maccabee of the Hasmoneans than the peaceful, love preaching rabbi of Christianity, Jesus of Nazareth. According to the Jewish sages, it would be by violent war and a heavy rod that this messianic figure was believed to restore the independence the Hasmoneans had lost to the Herods. Who was this figure?
The Jews believed this prince was to be Simon Bar Kokhba who was anointed as nasi or "prince" by the high prince after the Romans allowed the Jews to return to Jerusalem after the First Jewish Revolt. Due to a fatal mistake of interpreting Ezekiel's Temple prophecy as literal and claiming that the Messiah would come only when there was a need to rebuild it in fulfillment of this particular oracle, the Second Jewish or Bar Kokhba Revolt ended the Diaspora in 136 CE. Bar Kokhba was killed, Jews were slaughtered again, and Jews did not see their holy land again until after the Holocaust.
This caused a practical end to the Messianic hope in Judaism. While it became a part of Maimonides 13 articles of Faith, Judaism as a whole, which is not dogmatic, never universally adopted them. The Reform Movement, for example, the first Jewish denomination, denied any belief in a Messiah. Reconstructionist Jews do not believe in a personal Messiah either.
Eventually, the Messiah became a personal hope generally shared among some Orthodox groups, though again not universal among each and every Jew.
The reason Jews do not believe in the Trinity is because Christians have a different view on the Messiah altogether. They see the Messiah as a spiritually salvific figure and one with God. Except for mostly Unitarians and Jehovah's Witnesses, accepting Jesus as the Messiah means to accept Jesus as the Second Person of the Trinity. And this is the main reason Jews do not accept Jesus as the Messiah.
There isn't anything bad about Jesus to Jews, personally speaking. A rabbi who teaches Torah and tells Jews to stop teaching personal traditions and who performs miracles with the power of God as well as stands for justice--what is wrong with that? He even rises from the dead! The only wrong thing is that Christians say he is God.
So this is why the Jews don't believe in the Trinity. God is also not so much of a person or perhaps even that much of an entity in Judaism. In Christianity, the Bible's descriptions of God being a personal God are literal because Jesus said so. Therefore this makes the Trinity more of a reality for Christians.