Try the memory sticks the machine came with and of course, inspect all vents and heat sinks for cleanliness.
I have a friend with a July '06 iMac (core duo 1.83GHz). OS 10.6.8.
If she lets it put itself to sleep, everything is fine. If she uses the Apple menu item (or the power button) to put it to sleep, when she tries to wake it up, it just restarts instead.
We tried reinstalling the OS, that didn't help. There is just the keyboard, mouse, and a printer on the usb ports, no hub - no Ethernet either.
I discovered that setting the hibernation mode to 1 instead of 0 (so that the mac hibernates instead of just going to sleep) gets round this problem - the mac comes out of hibernation when woken up, instead of restarting, which at least saves the current session.
Any ideas as to what might be causing this restart problem? I'd really like to fix it.