Born in Fort Gaines, Georgia, Bryant studied at Abraham Baldwin Agricultural College. He was drafted in the first round of the 1981 amateur draft by the Los Angeles Dodgers, having previously been drafted by the Dodgers and Minnesota Twins, but did not sign with either team that year. Bryant was promoted to the major leagues for the first time in 1985, but was unable to establish himself as a regular outfielder, and spent the next three years traveling back and forth between the major and minor leagues.
In 1988, the Chunichi Dragons of the Japanese Central League showed interest in Bryant, and he was shipped to Japan in May, 1988. On June 7, 1988, Dick Davis, the cleanup batter of the Kintetsu Buffaloes, was arrested on drug charges, and the Buffaloes were forced to let go of one of their best hitters mid-season. Unable to fill the offensive void left by Davis, the Buffaloes found Bryant, who was playing on the Dragons minor league team. After seeing Bryant blast a home run in a minor league game, the Buffaloes offered to purchase his contract. The Dragons minor league staff was opposed to the trade, but regulations allowed for only two non-Japanese players per team, and the Dragons already had Taiwanese baseball star Genji Kaku and slugger Gary Rajsich on their roster, leaving no room for Bryant. The purchase was finalized on June 28, giving birth to one of the best left-handed power hitters in Japanese baseball history.