Is your system overclocked??
Have you tested the stability with Prime95 (recommended) or IntelBurnTest??
Does your new graphics card require its own separate voltage supply (it will have either one or two cable connectors on the card to which you should connect a separate feed from a special cable from the PSU).
Do you have a power saving application installed (Gigabyte boards have a utility called Dynamic Energy Saver or DES for short) - this reduced the voltage supplied to various components on the motherboard when it is not under load - could be that it is reducing the voltage too much and this is causing the crashes. If you have one of these utilities, then switch it off.