Go into the IP settings on your computer and write down the DNS
numbers that your working internet connection on your computer
sees. Then also check the WAN config page of your wireless
router, which you should be able to access from the computer
web browser, and see if the DNS numbers match on that page.
Then finally on your XBOX MS dashboard, goto until
the network settings, and manually config the DNS settings
to the ones you wrote down, or the ones from WAN config
page of your wireless router.
Also make sure you update to the latest dashboard, as the
moer recent ones like 5960, support "wireless" operation.