Enter codes at the Main Menu. To enter a code, pull the trigger multiple times depending on the position of the cheat code letter in the alphabet. Then press the Duck button (default on the Guncon is B) to enter a letter.

For example, to enter the code BLAM, pull the trigger 2 times for "B", press Duck, pull the trigger 12 times for "L", press Duck, pull the trigger 1 time for "A", press Duck, pull the trigger 13 times for "M", then press Duck. The screen will disappear and then reappear if you did it correctly.

BLAM - Unlock Arcade mode
ABROAD - Unlock Country Challenges
MIGHTIER - Unlock all of Mighty Joe Jupiter
LETSBOOGIE - Unlock the jukebox all all music tracks
MEBIGCHEAT - Unlock everything

For easy reference, here are numeric versions of the codes:
BLAM - 2, 12, 1, 13
ABROAD - 1, 2, 18, 15, 1, 4
MIGHTIER - 13, 9, 7, 8, 20, 9, 5, 18
LETSBOOGIE - 12, 5, 20, 19, 15, 15, 7, 9, 5
MEBIGCHEAT - 13, 5, 2, 9, 7, 3, 8, 5, 1, 20

Thanks to Revolution reader Trish!