I have seen quite a few posts about this but mostly with people saying to lower the settings in their config or disabling DWM. I have tried each of these options with no fix in sight.
Tonight I made a few changes with my settings and tested a few things and found one simple setting that solved my issue. Now I'm not guaranteeing this will work for everyone but it is certainly another option to add the the pile.
I have 2 Sapphire Radeon HD 6870's in X-fire. Plenty of power to handle the graphics I can assure you. The one setting that I changed that by default CCC had set was the Tessellation setting. I had it set to use the AMD optimized tessellation setting due to FFXIV's lack of graphic options. Upon changing this to "Application Controlled" I was able to get play throughout the Shroud for approximately 2 hours without a single line across my screen. I even tested this with my settings back maxed out again and not a single issue. Make sure if you are using a Radeon card that you have everything under the "Gaming > 3D Application Settings" set to 'Use application settings.' I also have my Catalyst AI set to Performance over quality.
I would say it never hurts to give anything a try at least once.
System Build:
CPU: i7 950 3.06Ghz OC to 4.0Ghz Corsair H70 Liquid Cooled
Memory: 12GB G Skill Ripjaw DDR3 1600Mhz
HD: 2x Velociraptor 10k RPM on Raid0
Video: 2x Sapphire AMD Radeon HD 6870 on Crossfire OC 1000/1250
PSU: 1000w Zalman
Config Settings:
Display Mode: Windowed and Full Screen
Window Size: 1680x1050
Multisampling: 8xQ MSAA
General Drawing Quality: 8 (Standard)
General Background Drawing Quality: 5 (High)
Shadow Quality: Lowest
Ambient Occlusion: No
Depth of Field: No
Texture Quality: High
Texture Filtering: Highest