ALL CARS
At the Main Menu, on the D Pad, press Up, Down, Right, Right, Left, Up, Up, Down. If you hear an engine, you entered the code correctly.

ALL TRACKS
At the Main Menu, on the D Pad, press Up, Up, Left, Down, Left, Right, Right, Up. If you hear an engine, you entered the code correctly.
Source: GameWinners

Championships:

STORY MODE
At the Main Menu, on the D Pad, press Down, Left, Up, Right, Right, Up, Down, Left. If you hear an engine, you entered the code correctly.

ARCADE CHAMPIONSHIPS
At the Main Menu, on the D Pad, press Right, Left, Up, Right, Down, Down, Left, Left. If you hear an engine, you entered the code correctly.

MULTIPLAYER CHAMPIONSHIPS
At the Main Menu, on the D Pad, press Left, Down, Left, Up, Right, Left, Down, Right. If you hear an engine, you entered the code correctly.