Cary first played with Joe Marsala in 1942, then played solo at Nick's in Greenwich Village in New York City in 1942-43. He also worked briefly with the Casa Loma Orchestra and Brad Gowans. During a stint in the Army in 1944-46, he managed to continue recording, with Muggsy Spanier and Wild Bill Davison among others. After his discharge he worked with Billy Butterfield, then joined Louis Armstrong's All-Stars in 1947-48. In 1949-50 he was in Jimmy Dorsey's orchestra, and in the 1950s worked with Eddie Condon, Pee Wee Russell, Max Kaminsky, Bud Freeman, Jimmy McPartland, and Bobby Hackett.
In 1959 he moved to Los Angeles, where he became an active freelance and studio musician. Beginning in the 1970s Dick led his own band, the Tuesday Night Friends, a group of top Los Angeles jazz musicians who enjoyed sight reading Dick's endless procession of original tunes and arrangements.