I'll recomend some exclusives and multiplats just in case you never played them (also a lot of suggestions from other posters are great like Conker, Jet Set Radio and Crimson Skies).
Brute Force - 4 player futuristic shooter, you can play solo and command the 4 characters at the same time and switch control between them has each as special abilities (one is a reptile and walks in four legs being the fastest)
Hunter: The Reckoning - This was also on the GC, its a top down shooter for up to 4 players where you kill monsters and shit, its a fun game overall (the sequel is Xbox exclusive though, while a different sequel is PS2 exclusive, what a great way to split up the fanbase).
Second Sight - Also on other platforms but that plot twist was amazing and I bet not many played.
Wreckless - The Xbox version is simply the best, its a fun game that honestly is only worth playing on the Xbox since the best part of it are the visuals.
Timesplitters 2 and 3 - The best multiplayer arcade shooters with the best campaigns (2 for challenge and variety, 3 for story and fun).
Blinx 1 and 2 - not the best games but they are somewhat different and you can see what was supposed to be Microsoft's mascot.
Dead or Alive Ultimate - Its Dead or Alive, the ultimate version of Dead or Alive basicly.
Fuzion Frenzy - A party game, not that bad but not that good, it does have an unique visual and there is a cheat that lets you play in first person view and understand how hard it is to see inside an hamster wheel.
Kung Fu Chaos - Great fighting party game, there is a lot of fun to be had with this one.
Midtown Madness 3 - Really loved this game, the only thing similar to it now is Driver San Francisco.
Project Gotham Racing - Very cool racing games but if you played the sequels I guess its less of the same.
Quantum Redshift - Its the Xbox version of Wipeout basicly.
Steel Battalion - Dont really expect to find this game or manage to play it but it is still a very special game.
Unreal Championship 2: The Liandri Conflict - Its Unreal with swords and shit, very cool.
Voodoo Vince - Never played it but from the people that played it they liked it a lot
Dino Crysis 3 - Meh, you get to fight cyborg zombie dinossaurs.
Tenchu: Wrath of Heaven - Never played it but it seems to be one of the best Tenchu games.
Ghost Recon 2 - A cool TPS, not all that tactical but still a fun game.
Rainbow Six 3 - Doesnt let you do the planning like on the PC but it has some exclusive levels and it plays great on the Xbox too.
Counter-Strike - Yeah, its Counter-Strike 1.6 on a console, its works and plays great though.
Sega Soccer Slam - An awesome soccer arcade game, really fucking good.
Here is a video with a lot of other games very worth a look:
Genocidicles said:
AFAIK, the original xbox is the only place to get Munch's Odyssey.
They are both on the PC and on Steam, dont know about their current state though.