Almost any traditional English song. The English had a tradition of unaccompanied singing. I suspect the reason why a lot of appalachian song work well unaccompanied is that many of them came from England originally.

I also find that the Bothy Ballads of North East Scotland work well unaccompanied. I suspect this is because that is how they were originally sung. Farm labourers generally did not have enough spare cash to spend on instruments.