If the radio is turning off and all the lights turn on/off on the dash, that means it is an electrical problem and not a mechanical problem. Since it is intermittent, it is going to be hard to track down. It doesn't sound like the dealer is trying to screw you, they are just being honest. As for paying for it, is it not in warranty?
At any rate, it sounds like something is loose or corroded. A few things I would try are as follows:
1) Turn the ignition into the on switch, but don't start the engine. Turn the radio up loud enough so that you can hear it from outside the car. Jiggle the battery cables and wiggle the connections on the battery.
2) With the ignition in the on position, try wiggling the key around. Also, try knocking on the steering column with your hand around the ignition area.
3) Wiggle the wires/connections around the interior and engine compartment fuse box. Also, tap with your hand on both of the fuse boxes.
If any of these trigger the radio to turn off and back on you've likely found your culprit. If not, you probably need to let the dealer do some intensive troubleshooting.