The first two rotating platforms you encounter are trying to tell you something. Listen to the beat of the background music to get an idea of the timing for when they rotate. This is a recurring obstacle throughout the level, so pay attention to when you choose to jump on those rotating platforms. For now you can land someplace safe; later on this will not be the case.
The first Puzzle Piece can be found above a series of three rotating wooden platforms. Above the third platform is a circular platform with grass that can be clung to wrapping around it. Get to the top of it to find a Barrel Cannon that takes you to a bonus chamber.
To the left of the first checkpoint is a pair of rotating platforms with rings of bananas over them. Collect both sets of bananas to reveal a Puzzle Piece above the platform furthest to the left. Make sure you hit the checkpoint first! It can be surprisingly tough to get those bananas.
Continuing from the checkpoint, you’ll find a small raised platform flanked on either side by a set of rotating platforms in the air. The third Puzzle Piece isn’t actually up in the air, though; it’s in a hidden room underneath the platform’s surface, concealed by leaves.
Not very far from the third Puzzle Piece is a chance to pick up the fourth, but it’s definitely more demanding. Three alternately rotating platforms bridge a chasm, with three Flaming Tiki Buzzes hovering between. You need to blow them out and stomp them while avoiding getting dumped into the pit below. Accomplish this and you’ll reveal the Puzzle Piece.
Eventually you will reach a large DK drum you can pound on. This breaks open the way to a rotating platform spanning a chasm between you and the second checkpoint. Cling to this rotating platform and get on top of it when it’s pointed vertical. The row of bananas at the top of the screen indicates another patch of grass you can cling to. Follow it and pound the gourd at the end for the fifth Puzzle Piece.
After climbing up a vertical path by clinging to several rotating platforms, check to the left to see a cluster of leaves obscuring another hidden ledge with bananas and the sixth Puzzle Piece.
The final Puzzle Piece is located at the end of another vertical climb up a series of rotating platforms. Look for a ledge with a yellow gourd at the end of it underneath the last of the rotating platforms, quite literally just before the Slot-Machine Barrel. Pound it open and reveal the Puzzle Piece!