Bit of a problem you’ve got there

What OS are you running on the machines, (if Linux you will need IP masquerading)?
How are they connecting to the router, via the Ethernet connection or wireless?

I take it you have the file and print enabled on all the machines.

-Make sure all your machines are getting their IP address from the router (set that up under TCP/IP protocol).
-Check the settings on your router, make sure that the time to live of an assigned IP address is round about 24hours is up to you.
-Make sure they are all set to go out through the router to the net.
-Try turning the firewall off on the router.
-Check that no firewall software on the computers is blocking packets from the router. (ICMP packets seem to be getting through but nothing else)
-Try turning off the wifi part of the router.
-Check what other network services you have running on your machines.

-If all else fails update the firmware on the router.

I had the same problem with my USR wifi router because it had the IP address 192.168.1.1 and a computer on the network had 192.168.0.1. Two machines fighting for default gateway, I could ping but not access files on other machines.

Hope this helps