I think you can use automated testing for games, usually... It requires a few compromises.

If you're feeding input into a non-deterministic game the character won't be exactly where they were last time, etc. You could keep track of the movement offset and see if it's within an acceptable range, or you could check the character's speed at two times and make sure they're accelerating properly, etc. Design a test level to highlight the potential problems (weird ground polys, whatever) and write an in-game script to test the behavior.

I think I could even script visual checks - mostly. You'd make, for example, a level prone to Z-buffering errors, where the colors were chosen to make it obvious - like a red wall showing through a white one. Save a stream of screenshots and compare them. Trivially, check for red. More complex, and perhaps better, check for high-contrast areas. It wouldn't tell you if the picture looked good overall but it'd be fairly good at finding instances of that one bug.

And for unit tests, don't be so quick to write such a good test that throwing it away is painful when you want to refactor the code.