My bad weather Univega Viva Sport has a set of those cheap gumwall-looking Kenda tires on it--I think they're K40s or something? I'm too lazy to look it up. I put them on last spring, and rode it a fair amount through the warm weather without noticing anything amiss. But I just tried riding it on rollers (getting cold here in Vermont), and holy cow! It was like riding a mechanical bull! Or maybe I mean a mechanical cow. Anyway, I checked and found that the back tire has a couple of flat/high spots that are maybe 1/8" from low to high. The wheel itself is round and true.
I've had a similar problem once or twice before when I'd done a sloppy job of mounting the tire, but in those cases letting the air out and pushing the bead down and reinflating solved the problem. In this case, though, that solution hasn't worked. Do I just need a new tire? I've run the same Kendas on other bikes with no problems, but I'd guess that defects are going to be somewhat common when you go for the cheap option.