Artifacting and crashing during games

I built my brother a new computer and we keep having problems with it artifacting and crashing during League of Legends.

Gigabyte Socket AM3/AMD 890GX/SATA3&USB3.0/A&V&GbE/ATX Motherboard GA-890GPA-UD3H
Asus Internal SATA 8x Blu-ray Combo BC-08B1LT
Thermaltake - 600W TR2 ATX Power Supply
OCZ OCZ3G1333LV6GK DDR3 PC3-10666 1333 MHz 6GB Gold XTC Triple Channel Kits
Phenom II X4 965 Black AM3 3.4Ghz 512KB 45NM 125W 4000MHZ
Sapphire Radeon HD 6850 1 GB GDDR5 VGA/DL-DVI-I/SL-DVI-D/HDMI/Display Port PCI-Express Video Card 100315L
Western Digital 1 TB Caviar Green SATA Intellipower 64 MB Cache Bulk/OEM Desktop Hard Drive WD10EARS


I've tested system using Prime95 torture testing and had no errors. I ran Memtest for 10 hours and had 0 errors. We've replaced the graphics card from a 5800 series to the new 6850. We also replaced the power supply with the 600 watt Thermaltake from a 500 watt. I don't know what else it can be...only happens during games. Of course we have the latest ATI drivers and all windows updates.