OK, well, I just figured this out. I did some web research and saw all sorts of suggestions for stuff like TSP, muriatic acid, and CLR (thanks, @mikes). But for the most part they're all bad for your lawn, bad for your skin, and bad for your wallet. Let me save you some money...
All you need is a stiff wire brush and a bucket of water (and the water is optional). You can even use one of those BBQ grill scrubbers (although I recommend a 1" wire brush). Just a little dab of water and a little elbow grease and the rust is gone! You might think of using a hose, but I wouldn't recommend it because, if you just fertilized your lawn, you run the risk of washing more fertilizer right back onto your cement, and you'll have the same problem all over again when it dries.
Now, I did this about an hour after I first noticed the stains. Perhaps if the stains had "set" for a few weeks it might have been harder -- I don't know. I did hear that another "solvent" that works well is vinegar (a natural acid), so if anyone doesn't get results from the above method, you might try it. But I didn't try it, so I don't know.