You're supposed to make it dizzy by rotating your sword in a circle (not a spin attack). Do it quickly, but not too quickly—if you go too fast, you'll start doing random swings, which won't work.
For the eyes that appear later in the temple, you need to have all of the eyes in a given room look at you at the same time in order to get past them. If not all of the eyes are open, they won't get dizzy and die no matter how fast you rotate your sword.
The First piece is located just to the left of the entrance to the Earth Temple. It's buried in a soft spot of soil, so you'll need to use the Digging Mitts to dig it up.
The Second Piece is located under the wooden watch tower down a path at the left of the temple. It's at the bottom of a hill of sand. You'll need to roll a bomb flower under it in order to have it fall over, so you can access the soft soil containing the piece below.
The Third Piece is located behind a boulder, about halfway up the giant sand hill in front of the temple (if my memory serves me correctly, it's how you got up to the earth temple to begin with). You'll need to roll a bomb flower in front of the boulder so it will destroy it, revealing a spot of soft soil containing the piece.
The Fourth Piece is a bit trickier to get to. You need to take the path to the right of the temple, and follow it along until you reach an area where it's too hot to stay long. Fi will give you a warning when you reach it, saying you can only stay here for a short while before dying. There are two ways out of this area, and you need to run for the correct one. Use your dowsing ability to find it. once you come out, you will be at the top of another giant hill.
Take the right path down this hill, and you'll eventually come across a ledge on the left wall of this path. You need to get up there, and then use the air geyser there to get across to the other side of the ledge, which will contain a soft spoil spot with the key piece.
For the Final Piece just proceed down the hill. Jump over the first platform, near the right, and then take another right. There will be a small cave containing a pool of lava, a boulder, a bomb flower and a sand hill. The key piece is located behind the boulder.
In order to break the boulder, you need to use a bomb flower rock, just outside the entrance to the cave, near the left. It will have lava leaking from it. Once you blow this up, the lava level will lower and you'll be able to roll a bomb flower over to the boulder so that it will be destroyed, opening the path to the bomb flower.
Finally, here's a video showing someone gathering all the pieces:
(Images taken from zeldadungeon.net)
Best Answer
According to several walkthroughs -- which I had consulted in desperation -- you have to crawl into that hole/tunnel because one of the pink switches is there.
I am 99.9% sure I never did that. I don't believe I even noticed the hole until after consulting the walkthroughs -- shame on me -- but at that point I had already raised the water level.
But as it turns out, I was able to move on. The water was high enough that I could jump to the vines on the right wall in the middle room and climb up. I thought I had tried this after raising the water level a 2nd time, but it's possible I didn't. (Or maybe I did but missed the vines because I was not pointed in quite the right direction.)
But I'm almost positive that I never crawled into that hole ... which would mean the walkthroughs are inaccurate on that point.