Square gem: use potion on a villager that makes you fell "light and springy" and put the villager under the cliff with the gem. the villager will spring up, snatch the gem, and bring it to the door.
Ash gem: once your fire has been burning for more than 10 hours, you will see sparkling on the fire. let the fire burn out, and in the asy remains there will be the gem. It will be to hot to touch at first (duh, it was in a fire!) so bring a villager to the water fall. the action should have something to do with putting water on it. repeat until the villager doesnt put any more on. drag a villager to the gem and they will put it in the door.
Clam gem: first you need to have re-built the statue. there will be a gem. give it to a master builder to "Finish" it, then give the finished gem to the cheif. the cheif will bring it to the clam and get the gem in the clam. they will bring that to the door.