There are 3 heart holders in the game. Their locations (also pictured below):
In a chest on top of a pillar near the start of Lilac Fields
In a chest in a cave in the forest where everything is dark
In a chest in an underwater cavern near the lighthouse
Simply avoid opening these chests and finish the game. The only real difficult parts without extra hearts are the Battle Tower and the final boss battle. I suggest bringing tons of potions wherever you go if you're trying to get this achievement.