These pieces all move exacly like their equivalent pieces in chess, with the exception of the cannon (which we'll come to later).
A brief note on naming: just as squares in chess are named a1, f3, and so on, squares in 3Dc are named Xa1, Yd7, etc., with X being the top board, Y the middle, and Z the bottom.
Back on the middle board..
The knight moves as per the usual game of chess. The other court pieces on the middle board, as well as moving as they usually do, move in the third dimension. Imagining a straight line from one square in any direction gives you twenty-six directions; along the three axes (6), at forty five degrees to each axis (12), and at forty five degrees to each of those lines (8). It's easier to imagine if you draw the boards, or if you hold one (real) chess board above another. If you've got a head for coordinates, then you should realise that square Yd4 is adjacent to Xd4, Xd5, Ze3, Zc4 and lots others.
In this manner, the king may move to one of twenty six squares, if it is in a central position. It can move to the eight squares as per normal chess, the nine squares above it, and the nine below it. Obviously, it may not move onto a board which does not exist.
The queen moves in exactly the same manner. Because the queen may move any distance, special care must be taken when moving from the top to bottom boards (or vice versa). If moving diagonally up (or down) two levels, it must take care to move two squares across in the appropriate direction. So Xb2-Zd4 is legal, but Zd4-Xe6 is not (it moved two boards up, one rank forward, and two files right; it must move two in every direction in which it is moving). It may, of course, move directly up, in which case there is no problem.
The bishop may move diagonally (as per usual), or diagonally and vertically. That is, starting in square Yd4, it may move to any of the squares Zc3, Zc5, Ze3, Ze5, Xc3, Xc5, Xe3, Xe5, or the squares to which it would normally move if playing chess on the Y board.
The rook moves, as you might expect, at forty five degrees to how the bishop moves. This allows the rooks to come into play much earlier in 3Dc than in normal chess, as it can move from, say, Ya1 to Za2 (assuming the pawn which belongs there has vacated the square), thence to Ya3 and a threatening position. It can also move directly up or down, to which the bishop has no analogue.
The only remaining piece to describe is the cannon. Just as the knight moves two spaces in one direction and then one space in a perpendicular direction, the cannon moves three spaces in one direction, two in a perpendicular direction, and one in a direction perpendicular to both previous steps. As there are a maximum of two boards vertically above (or below) the cannon, the first component of the move must be on the same board; but after that, the cannon may move two boards up and one square across, or two squares across and one board up. The cannon must obviously change board every time it is moved, and it moves a long way; it is the most common cause of anguish to beginner players. To combat players who are more experienced, many who have played only a few games initially attempt to take all their opponents' cannons, temporarily neglecting other aims in the game.
Special moves in the game are preserved. Every pawn may be promoted to any piece except a king or prince; en passant is a legal move. The pawn captures with a diagonal move but may only move forward when not capturing. The pawn moves 1 or 2 spaces, according to the normal chess rules, and cannot change levels (until it is promoted). Castling is legal, but only kings and rooks may castle (i.e. princes and galleys may not). Castling has the usual provisos (neither piece may have moved yet, and the king may never be in check while castling). Checking the king must be announced as usual.
The aim of 3Dc is also slightly more complex than in chess. Mating the opponent's king will win the game, as will capturing both his princes. You do not check a prince and thus do not have to declare when you threaten one. Forking the king and prince is a popular move, as the king must be moved out of check (of course, capturing the offending piece is a better solution..).