Hmmm...here's what I would check in approximately this order:
1. Before opening your G5, I'd check to make sure that the little "T" shaped Airport Extreme antenna is plugged into the back of the case.
2. Then I'd open your box and check to see if the coax cable is plugged into the little jack on the Airport card. This cable runs back through the box to the antenna jack I just checked in item #1.
3. Open your system preferences, then click on "Network".
4. Assuming you're using OS 10.3, you'll see the status of your various network connections. Check to see what it says for "Airport".
4a. If it's off (with a round red flag next to it), then turn it on (click "connect" at the bottom, then select the button to turn it on). Go to #5.
4b. If it has a yellow flag, it's turned on (recognized by your OS as well), but not connected to an access point. Go to #5.
4c. If it says that there is no Airport card, then you may want to try re-seating your card (with the power off!). Make sure you're inserting it the right way. It should snug down tightly for the last 1/4 inch or so of its travel as you seat it. A very slight tug outwards should not pull it back up again--it's supposed to be tight in its socket. Power up and go to #3.
5. Assuming all parts are plugged in, and your OS sees the card, then (with Network open from step #3) click on "Configure".
6. Open the tab for "Airport" and notice the item "By default, join" with a pulldown menu next to it. Click the menu item, and enter the exact name (called the SSID) of your router or access point (please tell me it's not "Linksys"--the default name from the factory).
7. Also click the "TCP/IP" tab and make sure it says you are using DHCP, unless you purposely and knowingly use static IP addresses, this should normally be the default.
8. Again assuming 10.3, use Finder to go to your main "Applications" folder, and click on "Internet Connect". Select the "Airport" tab. If you click on the "Network" button, it should show you the names of all the 802.11 access points it can see. Click on your system's SSID. I would also turn on the check box that says "Show airport status in menu bar" for future ease of use.
That should be it. Sorry if I've insulted your intelligence with anything here--just being thorough. Obviously you know a lot of this already, since you set up your laptop properly. If none of this works, you may have a bad Airport Extreme card (which is EXTREMELY rare, by the way). My guess is either that, or that step 1 or 2 or 4c will fix it. Some people miss the antenna connections AND adding the antenna on the back as well.
Hope this helps!
WCat