When playing a backwards-compatible original Xbox game on the Xbox 360, most (if not all) of the cheats for those original Xbox games will work just fine. The only tricky part is if the cheat calls for the use of the Black or White button.
This problem is easily solved by using the Left Bumper (LB) when the cheat calls for the White button, and the Right Bumper (RB) when the cheat calls for the Black button.
For example, if the original Xbox game cheat was Up, White, A, Down, Black, X, then you would enter that cheat when playing that game on the Xbox 360 as Up, LB, A, Down, RB, X. Happy legacy Xbox cheating!
Check out our illustrated guides on how to enter cheats on these video game systems: