Oblivion can crash a lot.
I'm assuming this is your only game that is a crash fest.
Lots of people seem to have problems with the game, there is a long list of stuff that helps.
Turn off ALL auto save stuff, on travel, sleep etc.
Anything that accesses your hard drive at the same time Oblivion does can cause a crash due to memory address problems. So if you have anything that auto checks for updates -reschedule.
Having the Pagefile on the same drive as Oblivion was a HUGE problem for me, it was the sole cause of 99% of my crashes.
Make sure you've patched the game with the official patch and the unofficial oblivion patch and look at any mods you've added, some people have terrible problems with replacement armor packs, others have none.
Oblivion is an intolleant hag for Overclocking for many people, try stock fsb, volts etc and see if that helps. You say you don't overclock but do you run default settings for each item?
On the hardware front, if this is happening at desktop and in other games then it points to hardware, check each item (memtest etc) individually where possible and see if you can find an incompattible or overheating item.