pHoaming Edison: "Wichita Lineman"
Combustible Edison: "Cry Me a River"
Judith Owen, Richard Thomspon and Debra Dobkin
Johnny Cash: "Hurt"
Fairport Convention: "Tried So Hard"
Eyrkah Badu and the Flaming Lips: "The First Time Ever I Saw Your Face" (video NSFW) and currently on auto-start at this link.
Amanda Palmer and the Flaming Lips: the replacement recording/video for the above (NSFW)
Gray Matter: "I Am the Walrus"
Bandits: "All Along the Watchtower"
Monsoon: "Tomorrow Never Knows"
The Slits: "I Heard It Through the Grapevine"
Fotheringay: "The Way I Feel" (preceded by a "bonus track" original: "Winter Winds")
Richard Thompson and Debra Dobkin: "Oops...I Did It Again"
...interesting to consider that the most popular versions of at least three of these were also covers, not including the Cash, where his version has all but eclipsed the original.