Well, it appears my router is dying - every 15 minutes, when I try to load a page, it says the connection was reset, and I have to restart the router to fix it. The weird thing is that downloads and stuff like instant messaging still continue to work despite not being able to load any web pages or start new things.
I tried resetting it to the factory defaults and even tried to use a proxy, with no luck.
So, I dug my old PowerMac G4 out of the closet and hooked it up to the modem. It's now configured as a wireless access point, and it's working pretty great. It's sad that that's about the only thing it's good for nowadays, as it still has the stock GPU and software vendors have dropped support for the PowerPC platform (like Mozilla and Opera, for example).
Going to get a new $20 router from WalMart soon, it should be a lot better. For one thing it will be a wireless N router, instead of the wireless G I currently have, so connection speeds should be faster.
BTW, it's a Linksys WRT54G, for those curious about the model of the dead router.