My HR34/700 has a similar issue. It will not show itself for about 45-minutes and then the picture will go black for a few seconds and then come back on for a few minutes and as time goes by it goes to black more frequently until it eventually goes to black for good.
I ruled out the HDMI ports on the Sony TV and ruled out the cable by using a different HDMI cable so the problem is with the Genie.
I have been using the component cables which work fine.
A reply to the original post indicates contact DirecTV so I guess that is the only answer.