There are 6 diamonds total in this stage. Some of them are in floating platforms that fall down when you jump on them so you'll have to be quick, however I believe most if not all of these platforms will resurface after a short time giving you another chance.
1. Head right from the start of the level and use the floating tubes or clam to jump across the river to the small island with the flip door on it. Before going in, jump on the platform to the right with the gold coins. Now you need to jump off, then back on the top of the platform as it falls to grab the diamond.
2. Jump on the floating tubes and use the to go around the right side. As you go you'll see this one on a set of platforms you need to jump up to grab the diamond.
3. After getting diamond 2, jump across left to another set of sinking platforms. You have to quickly scramble up to the clam shell (hold A to jump higher) and move right and FORWARD (away from the screen) onto the rock ledge above you - there's a small part you can land on. From there jump up to the top and head left.
4 and 5: These are both in the Yeti area that gets you the Someone Call For a Plumber? achievement. It's at the back of the level through a flip door near one of the audience members you need to save. One diamond is on the top right of the Yeti area, you can reach it with the clam shell. The other appears after you beat the Yeti.
6: The final diamond is at the highest point in the level, above the main stage. You need to go to the left first, then climb upwards (you have to go here to save some audience members anyways). Before or after you save the audience members by fighting the enemies here, climb up to the roof by using the ledges on the right side of the area.