Not exactly what you're asking for, but there's this thing called "shape note singing."
There's a long story of why it exists, but people still do it, and even though it has roots in religion, and the songs are religious, the groups you find today never perform in a religious capacity.
Get a buddy and go to one of the meetings. Everybody sits in chairs, in a circle and everybody sings. No instruments, just voices.
It's a beautiful, eerie sound which always falls flat in recordings.