I found this at the extremely valuable Folk Music Index:

Riley, the Furniture Man
Rt - Keno, the Rent Man

1.Double Decker String Band. Giddyap Napoleon, Fretless FR 144, LP (1980), B.01
2.Georgia Crackers. Poor Man, Rich Man: American Country Songs of Protest, Rounder 1026, LP (198?), cut# 4

The Georgia Crackers version would be the one I have, so now I know what I am looking for.

Following the link to Keno, the Rent Man turned up:
Keno, the Rent Man
Rt - Riley, the Furniture Man

Rm - Leaving Home

1.Cofer Brothers. Goin' Up Town. Old Time String Bands, Vol. 2, Marimac 9111, Cas (198?), cut#B.10
2.New Lost City Ramblers. New Lost City Ramblers, Vol. 4, Folkways FA 2399, LP (1962), cut# 14

So now, you know where to look, too, Pete!