Johnny Cash's cover of "Hurt"
Natalie Merchant's "Beloved Wife" and "Seven Years"
The Dixie Chicks' "Travelin' Soldier"
One of my favorites is It's a Wonderful Life by Sparkle Horse. I mean, sure, they steal a few lines from the Velvet Underground on the song but imitation is the sincerest form of flattery I guess. The music is slow, the lyrics are touching and the song can depress in no time flat.
A lot of what Mogwai has done is great ''sad'' stuff. This one: R U STILL IN 2 IT? Check the lyrics too. I like it particularly when the chorus comes in. ''Let's spend some money'' is what I tell myself when in sad mode sometimes courtesy of Mogwai :)