A few people are experiencing slowdown issues with patch 3.1, and since our last bit of technical help went over so well, we figured we'd do a little troubleshooting for you on this one, too. If your computer is running slower since you updated into patch 3.1, the first thing to do is check your video options -- Blizzard has tweaked a few things in there, and chances are that by tweaking them again, you might be able to fix your problem (or at least make it a little better). Specifically keep an eye out for the new Video Mode Ultra setting -- that specifically is not designed to be used unless your computer is current and top-of-the-line. Shadows also are quite a drag on the video card and CPU, and turning them down won't affect gameplay that much.
If your options are already low, the next thing you might do is check your videocard's make and driver version (scroll down to the "manually" section there -- you don't need to run Intel's program). Nvidia, who makes the common GeForce series of video cards just updated their drivers to version 182.50 on April 2nd, so if you have an earlier version than that, running the update will probably help. If you have an ATI card, you can find the drivers over on their site.
And of course if all of your software is set up and you're still having issues, there's always the possibility of updating your hardware. That can be quite an ordeal, though, so if you're not so sure on how to install new RAM or can't recognize the difference between SATA and IDE, you might want to enlist a friendly techie for a little help. WoW is still very forgiving, but Blizzard has been slowly adding on the graphical goodness, so if you've been playing with the same PC since launch four years ago, it might just be time for an upgrade.