Ok, after some vast and varied testing, I still haven't found the problem.
However, hopefully I have a little more information that might give someone an idea of what is wrong.
To recap, when I play various games (SW Battlefront, JK Jedi Academy, Far Cry), my computer will suddenly reset and load the boot sequence again. This seems to be a problem with my Radeon 9600XT, but not directly, maybe it's not compatible with something else in my system? Anyway, I swapped my card with an nVidia GeForce4 TI 4800 SE and the games ran fine, no reset. Incidentally, I put my Radeon into the similar machine I took the GeForce out of, and the games also ran fine again. So it appears that my graphics card is the cause of the problem, but it's not a fault in the card itself because it still works in other systems.
Now comes the odd part, I accidentally ruined the IDE cable connecting my two hard drives to the motherboard and so replaced it. Now, when I play Battlefront, instead of it resetting as soon as the level has loaded, I can actually get a fair bit of play (5-10 minutes) in before it resets. Would anyone have any idea why the IDE cable would have any affect on this? It makes no sense to me at all.
In any case, the problem still exists but it's not as severe. Also, I tried swapping my PSU with a 480W supply (which I tested in the other system which worked with my Radeon) and it makes no difference.
This is very baffling.
"Computers are useless - They can only give you answers."