Here are a few that I thought of off hand---
Parting Glass - The Wailin' Jennys
Fiona Apple - Hot Knife (It's not entirely without instruments, but the song is pretty bare for long lengths of it.)
Jane LeCroy (She sings most of her poems, so see what else she has out there.)
They have some similarities to the sample you provided, but let me know if you had something else in mind!