Definitely sounds like it COULD be the power supply. Although that would be pretty unlucky if the difference between the old CD drive and the new DVD drive was enough to push it past the tipping point.
One thing you might check -- does the system do this when the sata POWER cable is plugged in to the DVD drive but not the data cable? Or does it only happen when both are plugged in?
If it works with the data cable unplugged but fails when both are plugged, it would suggest to me that there's some kind of compatibility/BIOS/driver conflict going on and that's what you should look at first. If it happens regardless of the data cable, I'd start with the power supply.