What model of Linksys are you using? If its one of the WRT54 series, you may be able to find third-party
firmware, that gives the Linksys additional capabilities and *should* fixes the problem you are having.
I use a Linksys WRTSL54GS with the OpenWRT firmware (white russian) and it fixes the problem you are experiencing.
You get the random disconnects usually when there are too many open connections at one time, where the older open connections have not been cleared, as new connections are opened.