There's a couple of potential issues:
1) There may not be a keepalive option on your router, or the timeout may not be configurable (I'm not familiar with Linksys routers, so I don't know).
2) The problem may be at the client's side.
My home Netgear ADSL router drops any quiet connections after a matter of minutes, and it's not configurable so I have to run a spinner in the shell to keep the my SSH connection going. It's not pretty, but it works.