Kennedy appeared in many notable films from the early 1940s through mid-1960s, including High Sierra, Champion, They Died with Their Boots On, The Glass Menagerie, The Desperate Hours, Trial, Peyton Place, Some Came Running, A Summer Place, Elmer Gantry, The Man from Laramie, Barabbas, Lawrence of Arabia, Nevada Smith and Fantastic Voyage.

Of Kennedy's film work, he is perhaps best-remembered for his collaborations with director Anthony Mann and co-star James Stewart on Bend of the River (1952) and The Man from Laramie (1955), in both of which he played sympathetic villains.

He also enjoyed a distinguished stage career over the same period, receiving a Tony Award for his role of Biff Loman in Arthur Miller's Death of a Salesman (1949).

He inaugurated three other major characters in Miller plays: Chris Keller in All My Sons (1947), John Proctor in The Crucible (1953) and Walter Franz in The Price (1968).

On February 5, 1959, Kennedy appeared on the episode "Make It Look Good" of CBS's Dick Powell's Zane Grey Theatre.

[3] In 1949, Kennedy won a Tony Award for best supporting actor as Biff in Arthur Miller's Death of a Salesman at the Morosco Theatre.