He played college football as a defensive tackle for the Michigan Wolverines from 1976 to 1979 and was selected by the Cardinals in the first round (sixth overall pick) of the 1980 NFL draft.
[6][7] He recorded 30 sacks in 1983–84 and 37.5 in a 41-game stretch of the strike-shortened 1982 season and from 1983 to 1985, whose total was the second best in the National Football League for that period, behind Dexter Manley of the Washington Redskins.
[9] In 1984 Greer was named a second-team All-NFC selection by UPI and finished third in the league with 14 sacks while starting in 52 consecutive games.
[10] He played the final five games of the 1985 season with swelling in his right knee, and led his team in sacks the first six years in the league.
[8] He was expected to retire by the Cardinals staff prior to the start of the 1987 season, but he decided to return for a final year, despite doctors warnings about his knee.