There are 36 heart container pieces in this game.  Here is the complete list of their locations.

HCP = Heart Container Piece, AL=Adult Link only, YL=Young Link only, BL=Both Links

- Hyrule Field - Look for fenced-in grass on the path to Lake Hylia.  Throw a bomb into the center of the grass to unveil a hole.  Defeat the bush guy and he'll sell you a HCP for 10 rupees.

- Hyrule Field - Exit the Market as adult Link and turn right.  Keep walking in that general direction and you'll see a lone brown tree.  Bomb the base of it, fall down the hole, then use the Iron Boots to sink to the bottom of the pond and get the HCP

- Market - YL - In a house in the back alley at night you'll find a woman who lost her dog.  Go outside and find the dog (Look for a white one in the corner near the Bazaar).  Give the dog back to the woman and she'll give you a HCP.

- Market - YL - Go into the market at night to play the treasure chest game.  If you make it to the end you'll get a HCP.  If you have the Lens of Truth you can cheat by looking into the chests with it.

- Market - YL - After showing Zelda's letter to the guard in Kakariko Village, return to the market and the Bombchu Bowling game will be open.  Play to win a prize.  Randomly the prize will be a HCP.

- Lost Woods - YL - Enter the Lost Woods and go left from the first room.  Stand on the short stump and play Saria's Song and the Skull Kid on the big stump will give you a HCP.

- Lost Woods - YL - Go right from the first room in the Lost Woods and climb down the ladder to find two skull kids.  They'll want you to play songs with them.  Repeat whatever they play and after a few songs they'll give you a HCP.

- Lon Lon Ranch - YL - In the far southeast corner of the ranch is a stone tower.  Go inside and there is a bunch of crates.  Move them around to find a hole in the wall and crawl through  to get the HCP.

- Kakariko Village - YL - Climb to the top of Death Mountain and talk to the owl.  Have him take you back down to the village.  He'll place you on a roof.  Drop down off the roof on to the roof of the Chicken Women's pen and then walk into the hole on the side of the house there.  Find the HCP next to a cow.

- Kakariko Graveyard - YL - First in the day time watch the kid walk around the graveyard.  Make note of all the places he stops and waves his stick.  Then go outside and wait on the steps of Kakariko village until the skeletons FIRST pop out, then go immediately into town.  Talk to the gravedigger and he'll charge you 10 rupees a dig.  Have him dig in all the places where the kid waved his stick and one of them should be the HCP.