Have you tried reseating all the RAM and the graphics card - there is one thing I would target for sure. Take it out and let the onboard graphics work the monitor.
That is if you don't mind me saying so a fairly basic card too Steve I had one myself and I just upgraded to a NVidia GeForce GTX 650 and it is brilliant.
I had a fair few problems with the ATi stuff and ma not going back in any hurry. Just as a matter of interest have you done anything in the Catalyst Control Centre - I did find re downloading that helped my problem for while. AMD Catalyst
< this is for the 5000x series cards.