A different router would be my last resort. Can you confirm that the Internet connection in de modem get's offline or does it really concern your wireless connection or is a wired connection also disconnected from
the Internet?
PS: don't try them all at once. Try one, try for a while and when it does not improve, move on to the next option.

Wireless:
-Try different channels (start at 1, etc)
-Use laptop to spot other neighbouring wifi networks and see what channels they're on (you can even use a tool like Wireshark)
- Check wireless driver on laptop/wireless client (newer one)
- Isolate the wireles device that's disconnection (take all others offline)
Network:
- Turn off all safety features (WEP, AES etc) on router and see if that is improving connection
- Turn off firewall on DIR655
- Check internal network with : no double NAT or double DHCP (in other words: do modem ánd router use DHCP and Network Address Translating? If so, this can cause serious problems)
- Assign static IP to DIR655
- Set connection speed to fixed 100 Mbit