Mountain Lion Hangs on Early 2008 Mac Pro

I installed Mountain Lion 12A269 on a freshly formatted disk on my Early 2008 Mac Pro. I have 10 GB of memory.

Once or twice a day, the computer hangs. I can move the mouse cursor, but can't click on anything and the keyboard is inactive. The dock doesn't raise when I move the mouse over it. CMD-OPT-ESC doesn't bring up the force-quit window and none of the menus work. All I can do is hold the power button in to shut down and then restart the computer.

There's no kernel panic -- the computer just hangs.

The console shows two System Diagnostic Reports:

Finder_2012-08-28-093935_Disk.hang

RPDLAgentHelperE_2012-08-16-14738_Disk.crash

Since it just hung a few minutes ago (08/30/12) I don't know if either of these files are relevant. Neither one is helpful.

I'm looking for suggestions for fixing this problem or additional troubleshooting. The computer just hangs -- there's no consistent action to trigger it. It's already resulted in a lot of lost work and productivity.