Those specs tell me it is MacPro1,1 - why not look at System Profile.
Maybe the PSU has empacked dust. Or the inside of the processor(s) or the cpus need to clean the contact and reapply paste properly.
There are 4 ways to reset the SMC. The "power on while unplugged" being one. Yours has a physical SMC button too.
But I would start with Apple Hardware Test.
2008 uses 800MHz and only comes in a 4-core SP or 8-core DP of 2.8 or 3.0 or 3.2GHz
Disconnect all wired peripherals except keyboard, mouse, and monitor, if applicable. Launch the usual set of applications you use when you notice the problem.
Launch the Activity Monitor application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ If you’re running Mac OS X 10.7 or later, open LaunchPad. Click Utilities, then Activity Monitor in the page that opens.
Select the CPU tab.
Select All Processes from the menu in the toolbar, if not already selected.
Click the heading of the % CPU column in the process table to sort the entries by CPU usage. You may have to click it twice to get the highest value at the top. What is it, and what is the process? Also post the values for % User, % System, and % Idle at the bottom of the window.
Launch the Console application in the same way as above. Post the dozen or so most recent messages in the log — the text, please, not a screenshot.
I have a MacPro 3,1 (Mfg Sept 2008) which I have used from 18 to 20 hours a day since it was new. A month ago it started having noisy intake fan issues (the dual fans at the front of the computer). At first I thought it was a harddrive issue, but after replacing all my harddrives with Intel Sata 3 SSDs (Speed Demon 520s, which make NO NOISE at all) I realized it was Fan Noise (sounded like they were out of balance during a cold boot).
Well, to make a long story short, the problem appears to have been a nearly dead CR2032 battery. At least the problem has not reoccurred or recurred again since replacing the 3 1/2 year old lithiums. It is at least worth a try if your battery is several years old.