Defeat the four fiends to unlock each new dungeon. Each features random maps, exclusive items, and bosses from other Final Fantasy Games. I recommend beating all four fiends before venturing into the dungeons.

Lifespring Grotto - Defeat Kraken. Located near the canal by Melmond.
Earthgift Shrine - Defeat Lich. Located near Temple of Chaos.
Hellfire Chasm - Defeat Marilith. Located in the Dragon Islands.
Whisperwind Cove - Defeat Tiamat. Located in the rivers near the Ice Cave.