The lines all start at similar locations. This makes me think it's a memory access problem. I'm suspecting a cracked solder joint somewhere.
Fire up the computer, open the case, and grab the inside corner of the vidcard (ground yourself to the case first!) Gently flex the card a small amount. If the screen goes apeshit, you've got a physical problem with the card; a cracked joint or maybe a delaminated layer in the PCB.
If flexing the card had no effect, check for stray heatsink goop on the board, and double check physical contact between GPU and heatsink. (If GPU temps are good, this is almost certainly OK.)
You might also try pulling the card and cleaning the contacts with rubbing alcohol and a cloth. Again, observe static safety procedures!
Good luck.