Vic and Sade
Just downloaded two episodes of Vic and Sade, which ran from 1932 to 1946. According to Wikipedia, "Part of the magic of Vic and Sade is that all of the action, all of the people and all of the places in the town were created strictly through the dialogue. Listeners heard just the voices of the three, later four, principal speaking characters, embellished with very few sound effects." Sounds promising.