As other answers have suggested, this kind of error is most often caused by a physical hardware problem. In order to rule this out (and make sure you don't have to pay for a new graphics card), you need more proof than running other games at high settings.
One of the most common graphics card problems, which can vary from game to game (yes, even at high settings -- graphics drivers are fickle things) is overheating. Even if you are sure your computer is clean and the fan intakes/outtakes are not obstructed, you should still test the temperature of a few components while running the game and getting these artifacts. If you don't have a tool for monitoring temperature inside your computer, please download and install HWMonitor. If you find that temperatures on the CPU and GPU are within acceptable ranges and you're still getting graphical artifacts, you can move on to further tests.
It is important to update your graphics driver and verify the program files in Steam (after all, there may have been an update for this very problem). In general, though, be aware that the graphics engine in Civ 5 is not very optimized; it may stress your system a lot more than games with better effects, even at a low setting.
You have mentioned that you have a "pretty good computer", but specific part names (starting with your graphics card) would also help diagnose the issue. Especially if your graphics card is still within warranty, you may find that replacing it fixes the problem.