So for this achievement, you need to leave all three 'peaks' (meaning the top card in each pile) alone until the end. So clear the piles but do not touch the top card in each pile until all three are visible. Once they are, just clear them. They do not need to be cleared in order, they just need to be the last three cards you clear.
Obviously, this takes a little luck. It'll help tremendously if you still have a big deck left to draw from once you have those three cards left. But with a little luck and dedication, this achievement isn't too difficult.