That's interesting. I recently retired my Linksys BEFW11S4 and replaced it with a Netgear WGT624 (the 108 Mbps turbo version). I don't remotely connect to Linux boxes, but I stay on AIM sometimes for days at a time. I never experienced the problem you're having with either router. Since you have the same problem on both machines, I would guess the router or cable modem is the culprit. Since you also see it using different routers, I'd suspect the cable modem, personally. It's possibly not getting something from your router that your computer supplies.
Oooh! Oooh! Oooh! Here's a thought. You're getting a leased IP address from your cable provider (like most people). I wonder if the cable modem periodically pings back toward the router/computer to see if your system is still up. If it get's no reply, it might close your connection and give your IP address away. One thing that would probably be different is that your computer would respond to the ping by default, but by default, the router doesn't honor ping requests (at least my WGT624 doesn't). You have to set a checkbox to enable a response to pings from the Internet on the 614, I'll bet. It was called something odd on the Linksys. Anonymous requests or something like that. Also, set to not accept ping requests by default. (In general, you don't want to respond to pings since that lets the script kiddies know there's something there.)
The Linksys has some sort of idle timout and keep alive settings as well, but I think that would be off by default. You might try those as well.
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