Hi Wedge
Firstly have a good look at all the connectors just to make sure, see if you can monitor psu voltages and tempretures for any further clues. you might find it useful to run memtest to realy check the memory out. make sure that the cpu heatsink is makeing good contact
if nothing forth coming then take it down to a bare bones system, remove anything thats not essential so your left with memory keyboard mouse monitor and a disk drive. if that proves stable replace devices one by one until you hit trouble again.
if not stable at bare bones then you need to swap psu, disk drive and possibly memory. and vga card.
if nothing found there your looking at the mother board.
make sure that your psu is man enough modern vga cards, mother boards and cpus are pulling a fair wack and the psu needs to be a debatable ( i'm sure others will help with detail) minimum of 450watts
hope you find the problem in the earlier steps as we don't all have spare bits to swap in and out.
Good luck
Rick