It almost has to be a hardware problem... if more than one game crashes.
But you have performed the common tasks that would normally fix it. Something in the original setup is obviously the cause.
Run the free Memtest86 to rule out memory. It must run for four hours or 7 passes, which ever is longer. If you are using memory with "Value" in the name or description, that could be the cause... or if your old, original memory is part of your upgrade, that is a clue.
If the memory tests ok, tell us more.
These problems are usually:
Video graphics card has problems. This is the most common cause.
Video graphics driver is bad, and needs to be replaced.
Hard drive is bad. Replacements are inexpensive, and you can always use an extra.
CPU heatsink is not connected firmly or thermal paste has dried up.
CPU fan is failing.
Power Supply is inadequate. Not all new ones are good.
Any defective component in a PCI slot can cause intermittent problems.