During his career at Rice, Kinney, who was an imposing 6 ft 6 in (1.98 m), 215 pounds (98 kg) center and forward, was a two-time consensus All-American. He was a Second Team selection in 1941 and a First Team All-American in 1942. He joined the Fort Wayne Pistons of the National Basketball League (a precursor to the NBA) in 1945. On January 30, 1949, he was sold by the Pistons to the Boston Celtics and finished out the year with them. At the conclusion of the season, the BAA merged with some of the teams from the NBL to form the NBA. Kinney was retained for the 1949–50 season, which was the NBA's first, and therefore the Celtics' first in the league. In 60 games that year, Kinney scored 667 points (11.1 ppg). Kinney's NBA career ended after that season, but he played for the Anderson Packers of the NPBL in 1950–51. In 23 games he averaged 12.4 points. On November 19, 1950, he tied a Packers franchise record with 28 points in a 73–81 loss to the Louisville Alumnites.