Branyan began his professional career with the rookie-level Burlington Indians of the Appalachian League in . He batted .211 with 10 doubles, five home runs and 13 RBIs in 55 games. The next two seasons, and , Branyan played with the Class-A Columbus RedStixx of the South Atlantic League. He was named the South Atlantic League Most Valuable Player in 1996 after he set a league record for most home runs in a single season, with 40. The previous record was held by Mike Simms who hit 39 home runs in . He was second in home runs and ninth in RBIs (with 106) in all of minor league baseball in 1996. He was named the Indians' 1996 Minor League Player of the Year (receiving the Lou Boudreau Award).