A third-round pick of the Astros in 1968, the Fort Stockton native got his big league debut in 1971. Greif (pron "grife") appeared in seven games, including three starts, and posted a 1-1 record with a 5.06 ERA that year.

He was traded to the Padres after the season with two others for pitcher Dave A. Roberts. He pitched there for four seasons then was dealt to St. Louis. Overall, he compiled a 31-67 record with 19 saves and a 4.41 ERA.