Okay so there have been WAY WAY too many people creating threads OVER and OVER and OVER again about the same issues with ATI/AMD Graphics Cards.
1) Borg Cubes "Rubber-banding" in the War zone @ Ker'rat: REALLY ISN'T a graphics problem, it's actually a collision detection error with the cubes, repair hulks, and other objects.
SOLUTION: Suffer until Cryptic/PWE can get around to fixing it.
2) Blacked out textures: If your using a name brand computer you purchased and it's not brand new, don't go by whatever the sticker or the box said that your component specs are/where. If it's an older computer and especially one that's seen alot of use your RAM, Processor, Graphics card might not be actually running at what it used to be. Usually a 4gig RAM will be anywhere between 4-4.4gig actually, so most time you won't notice if it's not still at specs, but if it was only slightly above 4gig, it might be below 4gig now. This also applies to your processor and graphics card.
SOLUTION: Check your system info in the control panel and make sure everything really is above the min specs for the game. If it is then see Issue 3, if it isn't then time for some upgrading or getting a newer system.
3) Blacked out textures and other graphical errors that just started: ATI/AMD don't really believe in doing a hotfix if a minority of users are having an issue with their drivers because of a specific issue. Instead they will just release a new version of the drivers and leave it up to the users if they update to the new one or not. This can cause any number of problems that weren't there before.
SOLUTION: Revert back to the previous version that worked without any issues for you. Typically the most stable versions of the ATI/AMD drivers tends to be the X.1 releases. Now sometimes doing a simple roll back doesn't always work and you might have to do a bit more work. If the roll back falls to correct the problem then re-download the version that was stable for you, but don't install it yet. Go into your control panel and uninstall all components for the graphics card from there. Then when that is finished check your programs file folder to make sure the AMD/ATI folder is actually deleted. Restart the computer and then reinstall. Don't forget you're going to have to redo all your graphics settings.
4) Particle Effects (IE: vent warp plasma, etc) These types of effects are killer on a graphics card, probably the most resource draining of any graphics. Most common issue is graphical lag
Solution: lower particle effect options, get a newer graphics card. Now most people aren't going to want to do either of those, so don't worry there is a third option to try. Note this isn't always going to be 100% effective, but go into your advanced system settings and change the options for virtual memory. By default this is set to let windows manage it, but sometimes good old windows just doesn't keep up with doing this. If you have the hard drive room to spare you're best best is to set that at 10-20 gigs. Windows like to keep this around 4-5 unless you really consistently give your system a workout so your best off just setting it yourself and setting it high.
5) When in doubt remember to breath. It's more that likely something that's easily fixed and probably already listed here. if not check the forums BEFORE creating a new thread because chances are it's an issue that's already been solved.
PS: 6) IF you're using a custom built computer especially one you had a friend or someone build for you and you didn't build it yourself, check the specs on your graphics card for the min power required. If you're power supply isn't large enough to handle everything in your computer the graphics card is more often than not the one that suffers the most. Remember even if it's not a custom built computer if you have too many external devices plugged in and your power supply just barely meets the requirements for your graphics card, those extra devices plugged in may be taking away too much power from the rest of the system. Always try to go atleast 50-100 above what you actually need, and you'll never have this particular issue.
EDITED 3/25/2012: Both Cryptic and AMD have both stated that they are aware of current issues with the DX11 in game and with the 12.2/12.3 driver set respectively. Please just be patient with both companies as they try their best to make these changes in a timely fashion. In the mean time there are still the solutions posted here, and also by other users in this thread and many others. Good Luck and Happy Gaming.