He attended the University of Arizona. He was drafted by the Houston Astros on June 2, 1987 in the 37th round (953rd overall) of the amateur draft. Dickson did not sign with the team, opting to go to college and try to improve his draft prospects. On June 4, 1990, he was chosen in the first round of the amateur draft (23rd overall) by the Chicago Cubs, and signed with them seven days later.
He played in 11 minor league games and went 7-3 with a very low 0.94 ERA. He also recorded 111 strikeouts in 76⅓ innings. After this performance he was called up to the major leagues and made his debut as the Cubs' starting pitcher on August 9, 1990. The second-youngest player in the league at the time of his promotion, Dickson's big-league career was nonetheless short-lived. He went 0-3 with a 7.24 ERA in his three starts, precipitating his return to the AAA Iowa Cubs. His last major league appearance was August 18, 1990.
After he went back to the minor leagues, he was chosen by the American Association managers as the best pitching prospect and possessor of the best breaking ball in the league. His record was 4-3 and had a 2.86 ERA in his 15 starts. He led the league with 92 strikeouts in 91 innings before he was injured by a stress fracture in his right foot in June 1991. During the following off-season he had arm surgery. He returned to the minor leagues after the surgery and retired from baseball in 1995.