When you reach in to the middle of the board, for example in order to move a piece from 4e4 to 5d5, it is extremely awkward and there is great risk of knocking other pieces over. You cannot play a game in an hour, simply because you cannot physically make the moves that fast.
A minor problem is that the stack is too tall. The shape ought to be a perfect cube, but because chessboards have squares of a certain width, and chess pieces have a certain height, the stack of boards is always too tall; and the 3D directions therefore do not line up as they should.
In addition, you want the chessboards to be transparent, which means they must be custom-made from relatively expensive materials.
Well, in theory it is, since I haven't built one yet.
You will notice that there is one unsolved problem. Feel free to solve it!
Perhaps if you use really the really cheap chess sets whose pieces are unwighted hollow shells of plastic, and if you have some spiffy tools that allow you to carve up the pieces in some clever manner, you can make the pieces fit onto the string without hooks.
In any case, the idea is that the pieces should hook onto the strings, or clamp onto the strings, or something like; and the unsolved problem is how this should be accomplished.
Because the strings get moved around and then snap back into place, the attachment should be firm; but because you need to detach the piece quickly and reattach it to its new place, many ways of making it firm will be too firm.
As you reach in, the square 4e3 (for example) gets pushed aside by your hand, and some of the extra meter of string is pulled up from underneath the floor of the cube; when you pull your hand back out, the weight at the end of the e3 string takes up the slack and everything returns to its normal position.
This still takes a bit more care than moving on an ordinary two-dimensional board, but is a lot better than the usual "stack of boards" arrangement.
A one-meter cube of space for the playing field makes a very large and heavy "chessboard" that is a major piece of furniture! However, this allows you to use the most common sizes of standard cheap chess sets, allows you to use cheap and coarse materials to build the "chessboard", and forgives a few small errors in the positioning of the holes and eyehooks.
The large size also means that you may often need to push fewer strings out of the way to make a move.
The weights hang down almost to the ground, which unfortunately means that you cannot sit with your knees under the floor of the playing area; but I think it is important to be able to push a string quite a large distance. Even so, there is not enough slack to push the a1 string to h8.
A smaller, space-saving version might use springs and spools instead of the weights, but would be quite a bit more expensive. I don't know of any sort of elastic that would stretch far enough and isn't fragile.
Here is what these would be in 3 dimensions:
Each square is a separate piece of tinted glass, held in place by shiny brass supports.
Each rank of 64 squares (that is, 1a1 to 8a1 to 1h1 to 8h1) is a separate structure.
The 8 ranks are connected by hinges, and you can open the board like a book in order to make interior moves.
Each stack (for example, 1a1 to 8a1) is a separate structure.
To move from 4e4 to 5d5, you touch e4 on your touchpad-board; all the stacks that are in your way rise up into the superstructure, driven by silent electric motors. You take the piece off of 4e4, touch d5 on the pad, and stacks are raised or lowered as needed. You put the piece on 5d5, touch "finished", and all the stacks are lowered to the normal position.
Your graphics person will want to do a 3D isometric view of the board. I suppose it is necessary to have such a display in order to impress shoppers; however, this display is not suitable as the primary interface to the game.
A sketch of such a display can be found in Hans's page on the Kogbetliantz Game.
The first thing you might notice is that the 3D cube is shown by putting your viewpoint at a corner of the board. Personally, I always sit right in front of the board when I play Chess, so I would find this viewpoint a bit strange; perhaps you always sit at the corner of the board, and would find this viewpoint quite natural, but for me, the corner viewpoint is already a Bad Thing(tm).
In addition, the picture shows the primary difficulty of 3D chessboards, the fact that our view of, and our access to, the square e4 on the middle level is blocked by several other levels of boards and pieces. You can do a certain amount with transparency, but then you must consider the difficulty of selecting a specific piece or square easily, quickly, and accurately.
The solution I suggest is that you should be able to see a completely two-dimensional view of any selected 8x8 layer of the board, either from top down or front-to-back or side-to-side: top-down is the normal 2D Chess diagram, front-to-back shows you (for example) all the pieces on the first rank from 1a1 to 8a1 to 1h1 to 8h1; and side-to-side shows you all the pieces on the h-file from 1h1 to 8h1 to 1h8 to 8h8.
While viewing a layer, the up/down arrow keys should quickly take you to the next layer; and point-click selects a piece or destination. (Pointing to a square displays its name, of course).
In addition, there must be a 3D display with transparency, and highlighing (the layer that is 2D-selected must be very 3D-visible, and layers that are far from it should be nearly faded out). The player really needs this to be able to see the 3D moves.
Both these displays must be visible at the same time, so your art director will be unhappy: the pictures for the 3D pieces need to be small.
Expanding on these principles with some common sense detail should produce a usable 3D computer chessboard.