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?