From the start run to the corners of the room and grab the rings and then head to the raised blue room, which has bunch of easy-to-get rings. Once you've gotten those rings, jump out of the room on to the centre pillar of the main room and grab the rings there. Then go left or right into one of the side rooms and then turn left if you went left or right if you turned right into the isolated green room. Collect the rings there before turning back and hitting the spings at the other side of the blue corridor.
You should now be at the outdoor part of the stage. Run around the edges of the room, collecting the rings there before jumping between the platforms, being careful not to hit the water since that drains your time, and getting the rings there. Once you've obtained all the outside area's rings head to side opposite the way you came in and head to the blue room, grabbing rings along the way.
You should see a platform with a large amount of rings on it. Jump to it, avoiding the water again, and grab the rings. Go out the other side of the room and grab the rings on the way to the springs. Once outside again, go to the other side and run down the corridor, collecting rings, until you meet the green room you didn't go into before. Grab the rings here and you should have enough. If not, head back to the starting room and use the fans to get the remainder.
This is the way I tackle it and it works for me.