Speedie grew up in Utah, where he overcame Perthes Disease to become a standout as a hurdler on his high school track team and a halfback on the football team. He attended the University of Utah, where he continued to excel at track and football before entering the military in 1942 during World War II. He spent four years in the service before joining the Browns in 1946, where he played as an end opposite quarterback Otto Graham, fullback Marion Motley and fellow receiver Dante Lavelli. The Browns, a new team in the AAFC, won the league championship every year between 1946 and 1949. The Browns merged into the NFL in 1950 after the AAFC disbanded, and Speedie continued to succeed as the team won another league championship. After two more years with the Browns, however, Speedie left the team for the Western Interprovincial Football Union (WIFU) amid a conflict with Paul Brown, Cleveland's head coach. He played two full seasons in the WIFU and one game in a third season before leaving professional football.