Alan Arkin was definitely a folkie; in addition to The Tarriers, he did indeed perform with the Babysitters. (And he was also writing for science fiction magazines at the time.)

There was a notably lousy group called The Coachmen who did a fairly funny parody combining "Tom Dooley" and "MTA." And there were, and are, the 3Ds; their first album, now available on CD, was well-known poems (The Bells, Annabelle Lee, Richard Cory) set to folk melodies.