Gubicza was a member of Kansas City's 1984 American League Western Division winning team and as well as their 1985 World Series winning team. However, he didn't get a chance to pitch in the 1985 World Series as the Royals decided to go with a four-man starting rotation. He was elected to American League All-Star Teams in 1988 and 1989.
Gubicza finished 3rd in voting for the 1988 American LeagueCy Young Award after posting a 20–8 win–loss record in 35 games (all as a starter), 8 complete games, 4 shutouts, 2692⁄3 innings pitched, 237 hits allowed, 94 runs allowed (81 earned), 11 home runs allowed, 83 walks, 183 strikeouts, and a 2.70 ERA.
He led the American League in games started in 1989 (36) and 1995 (33).
Gubicza has served as an MLB Analyst on Dodgers and Angels pre-game and post-game shows for Fox Sports West/Prime Ticket. In 2007, he was paired with José Mota on fifty Angels telecasts on Fox Sports West/Prime Ticket and KCOP.  He also co-hosts Angels in the Infield with Bill Macdonald. The Angels announced that he will work roughly 75 games on television during the 2008 season. Gubizca was slated to work with Rory Markas on Angels broadcasts in 2010 before Markas's sudden death on January 4, 2010. On March 3, 2010, Victor Rojas was named as Markas' replacement.