Any form of warranty on that machine?
It could be the power supply, not easily changed or, as you said, the logic board.
If the latter, then a new machine makes more economical sense than fixing it.
If you have an Apple Authorized Service Provider, or an Apple Store, near you, it may be worth taking it in and getting them to diagnose the problem for you. It probably won't be free though.
Here's my computer : iMac late 2006, core duo 2.0ghz, 250gb hard drive, 2gb ram.
My issue is, it's been running great with no problems until this morning. Overnight it was left running (with the screen off) downloading some large files. I turned the screen back on today about 10am and all was well, until I loaded up iPhoto 09. On opening iPhoto, the spinning beachball popped up showing it was thinking, and then suddenly the iMac switched itself off! Obviously concerned, I tried to switch it back on, but no luck, noting happened. No fans spinning, no HD noise, no lights, absolutely no activity!
I checked the apple support website on my other computer, and did the fixes for the PRAM (take power cord out, then wait 15 seconds, then plug it back in holding the power switch) - this seemed to fix it, but it had the fans whirring like mad!
I checked further on the website, and saw the SMU controls the fans and how to reset it - did this - computer wouldn't boot up at all!
After some time trying to fix the problem with no luck, I took the RAM modules out and reinserted them. This booted the iMac up with no problems and no fans spinning like mad, but...1 of the 1gb dimms wasn't registering. I figured OK the ram chip is bad and that caused the problem, I'll get a replacement. However, not long after the iMac did the same - turned off and won't turn back on. I tried the same trick but it wouldn't boot. After a long time messing around with the PRAM/SMU/Power cord I finally got it to boot and did a memtest on the remaining Ram Dimm - no problems.
Still however the iMac is not working right and just randomly seems to turn off/won't turn back on. I checked the temps/fan speeds etc, nothing wrong there.
Anyone have any thoughts on what this could be? Is it bad ram chips? Is it the power supply? Is it the logic board?
I checked the power cable/plug etc, they are all working with no problems at all.