Since December 2007 the Xbox 360 has been capable of playing Xvid files. But it's ridiculously complex to set up.
First of all: do not use Windows Media Center. Do not use the "Media Center" option in the Xbox. Do not set up anything called an "Extender". Windows Media Center is interesting and has a fancy UI, but for some baffling reason it does not support DivX/XviD.
Instead you need to do something simpler; just let the Xbox find the AVI file and play it. The easiest way to do this is via an external USB drive; just plug it into the Xbox 360 (it has USB ports!), go to Media/Video, press X to change source, and browse the files on the drive. It's really that easy. But copying files to a drive is a drag, what you want is to serve them over the network. And for that you need a UPnP server, a simple file sharing protocol.
There's a slew of UPnP servers out there for Linux and near as I can tell none of them work. Windows has a UPnP server built into recent versions of Windows Media Player. Launch WMP, click the obscure little arrow below the word "Library", and choose "Media Sharing". If your Xbox 360 is turned on you should see it as a device option. Allow sharing to it and you're done. You'll probably want to also do "Add to Library" to pick the folder where all your videos are. Then on the Xbox go to Media / Videos, press X to change source, and you should see your new UPnP server. Now you can browse and play.
This sharing process is all ridiculously complicated and obscure. Poor Microsoft; they really want to sell media devices, competing with iPod and iTunes and Apple TV. But it's clear their media stuff is designed by a giant committee: they lack the single person with good taste and the authority to say "make this work simply".