Poche was drafted by the Angels in the second round of the 1986 draft. He made his major league debut with the Angels in 1989, and spent most of the next three seasons with the team. In 1992, he was traded to the Reds, where he spent the next four seasons. He finished his major league career with the Padres in 1996.
Overall, Poche posted a 53–52 record with a 4.26 ERA in 157 career games. He struck out 535 batters in 923 innings pitched.
After retiring from baseball, Poche became a pitching coach. He has worked in the minor league systems of the Angels, Reds, and Seattle Mariners. He is currently the pitching coach for the Louisville Bats, the Triple-A affiliate of the Cincinnati Reds.