Burnout 3 will run on the 360, Burnout 4 for the 360 actually fixes the problems they had with the regular Xbox and PS2 versions, so 4 (aka Revenge) is a better bet for 360 owners, its purdy.
Xbox 360 specific games:
Action/GTA style:
Crackdown: good, but a little short
The Outfit: completely worth 15-20 bucks, a little campy, ww2 action, good coop.
Lost Planet: I liked the demo, but the reviews fell short of the exp[ectations set by it, so I never played more than the demo
Lots of people liked THQ's SAint's Row, the demo was meh for me, but multiplayer was supported and supposedly quite good.
Shooters worth owning:
Gears of War - Between this and Fight Night (below) one of my all time favorites on the system so far.
Ghost Recon Advanced Warfighter (#2 is supposed to be excellent, not tried it yet)
Rainbow Six Vegas (loved it, multi is great, single player is good, and its purdy, I think Gears is a better game, but, R6 is tasty)
Call of Duty 2 (and when you finish, trade it in and get CoD 3, 3 has much better multiplayer support).
Quake 4 is probably worth owning for 20 bucks if you can find a multiplayer game, great nostalgia, collectors edition comes with full version of Quake 2.
Wife/girlfriend friendly game:
Viva Pinata (drug trip gardening game with pinatas)
Hexic (puzzler, comes free on the HDD)
Racing games:
Need for Speed Most Wanted
Test Drive Unlimited (possibly my favorite it does

of things right)
Burnout Revenge (aka Burnout 4)
Project Gotham Racing
I have heard good things about Moto GP, but earlier versions were too hard for me, so I wimped out.
RPG:
Phantasy Star Universe (has a decent singple player campaign, very jp, and its online is an MMO-lite)
Final Fantasy XI (if you like MMO)
Oblivion (if you haven't played it on the PC)
Fighting games:
Fight Night Round 3 - possibly the best fighting game i have ever played.
DoA 4- Meh, I traded it in, I was very dissapointed with it and I loved earlier installments, your mileage may vary.
And my brain has stopped funtioning, I am trying to picture my game shelf with the 260 games in it, and blanking out a coup0le of good ones, so I can post more when i get home.

of the "must have" Xbox 1 games are on the BCL though, so the library is pretty decent and varied. Halo 1 and 2, Crimson Skies, Star Wars Battlefront (taken as a multiplayer game only), The KOTOR series. Xbox Live Market place also has some great gems, ranging from classic to new.