Grissom played baseball at Florida A&M University, and in 1988, the Montreal Expos selected him with the 76th overall pick in the June draft, as part of that draft's third round. He had been considered a prospect as both a pitcher and an outfielder, but the Expos decided to have him abandon the mound and work solely as a position player. He made his professional debut with the Jamestown Expos of the New York-Penn League that fall and advanced quickly through the system, first appearing in the majors on August 22, 1989. He showed steady improvement for the next few seasons, gradually developing into a star as Montreal's leadoff hitter and center fielder. He led the National League in stolen bases in 1991 and 1992, was a member of the NL All-Star team in 1993 and 1994, and won four consecuti