Linton was drafted by the Toronto Blue Jays in the 43rd round of the 1986 Major League Baseball draft, signing with the Blue Jays on September 5, 1986. He made his professional baseball debut with the Myrtle Beach Blue Jays of the South Atlantic League in 1987, where he had a very impressive season, earning a 14–2 record and a 1.73 ERA in 21 games. He was promoted to the Knoxville Blue Jays of the Southern League late in the year. He pitched in only one game for Knoxville, allowing three runs in three innings.