After they choose three cards, lay out three stacks from right to left, face down. Deal 10 cards to the stack on the right, and 15 cards each to the next two stacks. You should have 9 cards left in your hand.
Have them place one of their cards face down on the stack of 10, then have them cut the stack of 15 to the left and place it on top of their card. Have them place their second card face down on the middle stack and cut the stack of 15 to the left, placing the top half on their second card. Finally, have them place their third card on the leftmost stack and place the nine cards in your hand on top.
Gather the stacks up from left to right, putting the left stack on top of the right. Move four cards from the top to the bottom, then deal out the entire deck, alternating between face up and face down. Ask them to watch for their card.
They'll never see it, but keep dealing the deck out face up, face down. Each time through the deck, pick up the face down cards and deal them out again. You can act confused if you'd like. Finally, you'll have just three cards left that are face down. Flip them all over to reveal the three cards they chose.