Here is the full list of all 23 3D Chess achievements.
You solved all puzzles
You solved all "Set Checkmate" puzzles
You solved 20 "Set Checkmate" puzzles
You solved all "Avoid Check" puzzles
You solved 15 "Avoid Check" puzzles
You solved 10 "Avoid Check" puzzles
You created your first puzzle
You solved all "Set Check" puzzles
You solved 15 "Set Checkmate" puzzles
You solved 15 "Set Check" puzzles
You solved 10 "Set Check" puzzles
You solved 10 "Set Checkmate" puzzles
You have beaten the computer on Hard
You have beaten the computer on Medium
You have beaten the computer on Easy
You played a Multiplayer-Game
You have played the Chaos-Mode
You solved 5 "Avoid Check" puzzles
You solved 5 "Set Check" puzzles
You solved 5 "Set Checkmate" puzzles
You solved your first puzzle
You have read the credits
You have read the Chessopedia