Your problem likely has to do with your video card's power management features. What it's doing is downclocking your GPU's memory when it isn't under load. With the AMD cards under some driver revisions, they clock the memory too low in low power mode, and that causes flickering. To fix this you can try updating to the latest driver revision if you haven't already. If that doesn't work, download Sapphire Trixx or MSI Afterburner, and look for the option to disable ULPS. That will disable the power saving feature that downclocks the graphics memory, and should eliminate the flickering.
If none of that works, you could have a bad video cable, a bad DVI/HDMI/Display port, or an issue with your monitor. To test for that you can always try plugging the monitor cable into a different port, or try a different cable. If you have another computer, try hooking it up to the display in question, and see if it flickers at 1080p.