I'm happily running 4 rigs all with 4x 7970s (16x7970s total) without powered risers and they run for days on end. They only stop when i decide to "fettle". I keep them cool though - all temps under 65 except one that runs hot for some reason with a high of 75. Could it be a heat issue?
Also what PSU are you using as it may be that it's wobbling under the load?
I've tried a Seasonic 1250w and a Rosewill Lightning 1300w, with similar results.
I have cgminer set to auto-fan with a target temp of 75 degrees. They run at 72 an ~50% fan.
I haven't considered that it can be a temp issue. I ran a single card like this for an entire week without issue.