What they have in common is that they are played on boards smaller than 8x8x8.
Let's say e1 is on top of a1, so we know which way is up.
It uses the normal set and board, and although you have to remember the up-down mapping, one or two games should be enough practice for you to get used to it. Thinking in 3D will still be hard, of course.
The Bishops are too short.
The most common error is wanting to move across the invisible borderline, for example Qd1-h5.
When you use the old-fashioned "abstract" translation to derive your 3D rules of movement, there's a 3D "diagonal" that isn't part of the game: imagine a White K on a1, a Black King on f2 (that is, one square above b2). In this position, White is stalemated.
So the first new rule is that if you're stalemated you lose, and the second new rule is that Kings may not make up/down moves except to capture (and Black's King starts the game at d8).
This is marvelous because King versus King is usually *not* a drawn endgame! One side always loses if the Kings are on different levels, and you need to figure it out long before you trade that last piece!
Perhaps Kings can use this extra dimension as well; I think they should be forbidden to.
This allows the Bishops to make nice long moves, but makes it very hard to checkmate. (And so perhaps billiards is better than cylindrical, unless you forbid the King to use wraparound moves.)
When a Bishop moves Northwest from e1, it gets to choose either d2 or h2 as its next square; from d2 it could continue to a5 and then d6 to b8; from h2 it could continue to e5 and then either b6 to d8 or h6 to f8!
Of course, the "best" 3D chess would logically be 8x8x8, but that is difficult in many ways.
In order to fix the short-length problem of the Bishops, make it a billiards board. Getting used to the 3D bounces might be tricky.
To bring it back to 64 squares, cut out the middle square of the 3x3. This looks a lot like the Cylindrical Chess board, but it's different because the Knight can jump over the hole.
Mapping this to the normal board, we get something like: a1 on the middle level, and under b1; b1, c1, d1, top level; e1 under d1; and so on. (I'm far from certain that this is the best mapping).
Using a physical board, I would prefer to stack 8 of the 3x3 doughnuts on top of each other and have the players move up and down. The reason is that it's hard to reach into the middle of a wide 3D board.
You could use a similar trick with a 4x4x8 board, of course; on each 4x4, 8 squares would be missing. Different squares might be missing on different levels! (This would create interesting terrain.)
Rooks, Bishops, and Queens in the lounge would be near the center but would have very limited mobility; Pawns would be unable to advance and promote; Kings might seek deceptive safety here, and Knights would be okay (although even their mobility would be limited).
It seems like the lounge is a place where pieces get some safety in the center, but at the cost of some mobility.
This would work, of course; the only problem would be reaching into the center of the lower board when you needed to make moves there.
You could also extend this idea by adding a submarine board.
You could also cut it back a bit, and have only a few bomber pieces and just a partial board on the upper level. For example,
Each move, each player may either make a normal move on the normal board, move a CloudHopper from one cloud to another (possibly capturing an enemy CloudHopper), drop a bomb from a CloudHopper onto an enemy piece directly below (thereby removing that piece from the board), or move an empty cloud (but if you have no more CloudHoppers, you may not move clouds).
Clouds are grey, neither White nor Black, and can be moved by both sides now, except the one that was most recently moved, which is said to have a silver lining (and so it is marked by putting a silver coin on it). One exception: if you move a cloud, move a CloudHopper onto it, and then hey you get offa that cloud, it becomes the little white cloud that cried, and can move again.
Clouds move like Kings, but do not capture each other; and of course only one cloud may be above each square.
Cloudhoppers move like Knights, but only from one cloud to another. CloudHoppers may capture each other.
White's CloudHoppers start the game above b1 and g1; Black's Cloudhoppers start above b8 and g8.
Clouds start the game above b1, g1, b8, g8, e4, e5, d4, d5.
The built-in strategic tension is that in an open game, there may not be time to use the CloudHoppers, but in a closed game, they can be very important.
This game is only slightly 3-dimensional, is certain to be a good game and easy for chessplayers to play, and building the extra equipment wouldn't be very hard.
A silly variant: squares under clouds are wet and slippery.
Unfortunately, the aerial part of Cloudhopper Chess isn't very 3-dimensional. Of course, this happened because I wanted a game for which it would be easy to make physical equipment, but wouldn't it be nice to have a game with more 3D?.
Above the ground there are 3 levels of airspace, in which two kinds of unit may be found.
Each turn, you can make either a ground move or an air move.
The Bomber cannot attack other aerial pieces.
In order to capture, the Bomber must start from the topmost aerial level; it descends in one fell stoop to the botommost aerial level, diving straight down, and kills the enemy piece on the square just below it.
Of course, before it can capture again, it must climb to the topmost level.
Each side starts with two Bombers, on the lowest aerial level, above the King and Queen.
The Interceptor cannot attack ground forces.
The Interceptor can destroy enemy aerial units by moving into the same square.
Each side starts with two Interceptors, on the lowest aerial level, above the Rooks.
The strength of the air units is limited so that they will not be overly important. In fact, an exciting gambit is to ignore the enemy air force; the tempi "lost" by aerial moves may allow a winning attack on the ground.
Oddly enough, bombers defend against bombers. A bomber directly below an enemy bomber prevents the fatal stoop. Of course, this is a defensive posture, and gives up the initiative.
I deliberately chose not to have helicopters, paratroopers, anti-aircraft fire, or reinforcements.
The basic principle of fighters and bombers in a game like this must be that fighters are faster; a basic principle of early aerial combat is that an advantage in altitude is precious.