Andy was signed as a second-round pick in the 1983 amateur draft, by the Cleveland Indians and scout Bobby Malkmus.

[1][2] Allanson would break into the Major Leagues on April 7, 1986, starting Opening Day behind the plate for the Indians.

The Indians, feeling they were perhaps a better-hitting, veteran catcher away from contention, signed Rick Dempsey for 1987, demoting Allanson.

He was picked up by the Kansas City Royals in January 1991, and was traded to the Detroit Tigers before the season started.

Allanson played 60 games for the Tigers, backing up Mickey Tettleton behind the plate, and became a free agent after the season.