There is a problem it looks like when the router is renewing it's IP, the DNS is either not updating or is updating to all 0's. I've had this problem before. You can verify this by waiting until the problem happens, then checking the status page of the router to see what the router has for DNS. Run an ipconfig/all to match the DNS servers on your computer to what linksys has. They should all match.
If you can't ping your ISP's DNS, maybe their public DNS servers are on DHCP addresses themselves. That would mean that when you obtain an especially long lease from your ISP's DNS server and during that lease the address of the DNS server(s) change, and a new DHCP update is not set, it would explain the problem. Verify the IPs of the DNS servers over some time.
If all else fails, maybe your ISP simply has flakey DNS servers and you should consider using some others -- there are public ones (but don't use the root). I run my own DNS server and this reduces browsing time.