The goal of the Three Heart Challenge is to complete a Zelda game without picking up any Heart Containers or collecting enough Heart Pieces to form a new heart container. Hence, all the Heart Containers you'll get are the original three (in most games) you start out with. This is able to be done in twelve of the fourteen Zelda games. The exceptions are Four Swords and Four Swords Adventure for obvious reasons, and Link's Awakening and Link to the Past, since you are have to collect the Heart Containers dropped by bosses to continue.
Legend of Zelda
Difficult, but possible!
Adventure of Link
In order to do the Three Heart Challenge in Adventure of Link, just never upgrade your Life stat.
Ocarina of TIme
Very basic to do. Probaly one of the most completed Three Heart Challenges.
Majora's Mask
Hard, but not impossible.
Oracle of Ages
Possible but quite difficult.
Oracle of Seasons
Possible but extremely difficult. This is most likely the hardest that allows this challenge. You have to be an very seasoned Zelda player who takes on these type of challenges with ease in most or extremely gifted to do this.
Wind Waker
Minish Cap
Twilight Princess
Phantom Hourglass
Spirit Tracks
Again, pretty basic.
Skyward Sword
For Skyward Sword, the same rules apply. However, it is more like a Six Heart Challenge since you start off with six Heart Containers, instead of the traditional three.