I did some more poking around and rebooted. I then saw that cgminer is running happily in the processes list, but after some time it disappears from the list

is it maybe not able to talk to the ASICs? All of the cables appear to be attached properly and I'm using a Corsair CX600 PSU so it should have enough heft to run it.
I'm REALLY confused!

Do you have configured more than one pool ? Maybe it is unable to talk with pool and then process stops. But it will restart every 3 minutes.
Try to setup failover pools and then you will see if cgminer walk to another pool and hashes there. You can see this in
Miner status under LSTime.
The 600W should be fine even for "normal" OC-ing.
