Here is the full list of all 53 MOAI: Build Your Dream achievements.
For completing seven levels in a row without chopping down a single tree
For removing 125 piles of stone
For scoring 200,000 points
For killing 200 ghosts
For removing 50 piles of stone
For killing 100 ghosts
For chopping down 50 trees
For gathering 1,000 units of food from the pond
For scoring 100,000 points
For scoring 65,000 points
For gathering 250 units of food from the pond
For scoring 35,000 points
For creating a character
For scoring 350,000 points
For scoring 500,000 points
For scoring 650,000 points
For scoring 800,000 points
For scoring 1 million points
Gift of the high chief to exiled rebels
Award from the council of elders
Gift of the high priest Balm-in-Cense
Gift from the Northern tribe's chief
Award from the council of elders and all islanders
For building 200 buildings
For gathering more than 5,000 units of food while playing a level
For gathering more than 5,000 units of stone while playing a level
For gathering more than 1,500 units of sandalwood while playing a level
For gathering more than 10,000 units of mana while playing a level
For gathering more than 3,500 units of wood while playing a level
For preparing 100 fields
For restoring 100 quarries
For planting 75 groves
For restoring 30 markets
For recruiting 500 workers
For building 20 temples
For healing 100 workers
For killing 500 ghosts
For chopping down 150 trees
For chopping down 300 trees
For removing 250 piles of stone
For gathering 2,000 units of food from the pond
For completing 20 levels without chopping down a single tree
For completing 40 different levels without killing a single ghost with your own hand
For completing 25 levels in a row without killing a single ghost with your own hand
For opening 250 chests
For repairing 20 bridges
For repairing 100 buildings
For completing every level of the Eastern tribe with three gold coins
For completing every level of the Southern tribe with three gold coins
For completing every level of the Western tribe with three gold coins
For completing every level of the Northern tribe with three gold coins
For completing every level with three gold coins
For earning every achievement