Kennedy got his break when he was discovered by James Cagney. His first role was of Cagney's younger brother in City for Conquest in 1940. He portrayed good guys and bad guys equally, appearing in Western films and police dramas.
He starred in several well-received films in the late 1940s and the 1950s, including Boomerang, Champion, The Glass Menagerie, Bright Victory, Bend of the River, The Lusty Men, Rancho Notorious, The Desperate Hours, The Man From Laramie, The Naked Dawn, Trial, Peyton Place, Some Came Running, A Summer Place and Elmer Gantry.