Recently I've been tinkering around with my home network as we've just added a third COMPUTER, and in my research on how to do it, I've come across the information that a router has a built in mechanical firewall. My question is, is the router firewall good enough for me to eliminate the other firewalls on my individual computers. Because each computer has different operating systems, I have three different firewalls operating. (Win XP - Norton 360, Win 2000 - Zone Alarm, and Win 98 - Outpost).

My router is a D-Link DI 604, and all computers link to the net through it.

That depends on what level of protection you need.
A router firewall will only provide inbound protection. I.e. protection against something trying to reach your computers from outside your network.
It will not be able to block malicious software installed on your computers that might be trying to phone home.
So if you're fairly confident no one with access to your computers or your network might accidentally INSTALL something nasty, a router firewall might be enough.

Personally I use both. The router provides a great first line of defense while my software firewall MAKES sure I have complete CONTROL over what programs are allowed to access the internet. It also protects my computer in case a computer infected with something is connected to my network. Not all my mates are computer SAVVY so you never know what might be on their computers.