For the longest time I had this constant problem with my router, a Linksys BEFSR81 (the 8-port version of the BEFSR41) constantly disconnecting the computers connected to it. It wasn’t the Internet connection going down, since I had a computer that was directly connected to the modem as well, and it never had any problems.
So of course when troubleshooting something like a router, you go and get the latest firmware. No luck there. I also tried resetting the router, restoring default settings, changing the ports used on the back, and so on.
Finally, I tried giving one computer DMZ access (I’m not forgetting about the security concerns raised by doing this). It worked. No more disconnections for that computer, and for now, I’m satisfied.
Posting this in case it comes in handy for anyone else experiencing disconnection problems.
[2008 Addendum: The only real solution to this problem is to get a better router. The BEFSR81 is at least 7 years old now.]