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Now to weave a few threads together and hope it doesn't tangle too much. 

Hi, Ben, Daniil, glad to see you here. Welcome to the intellectual
free-for-all. I am most happy to have 4D and 6D represented. Off the top of
my head, I only know of VR Parton, LL Smith, and Dan Troyka doing 6D
variants, and there is a headline article in a recent Variant Chess
magazine where I believe the editor, John Beasley, analyzes a 2x2x2x2x2x2
game. Oh, and Larry will deny it. ;-) He doesn't believe in more than 3D.
He sees higher than 3D boards as nothing more than complicated 3D boards.
Didn't stop him from designing the darnedest 6D board - it has a hole
right through the middle of the 4th, 5th, and 6th dimensions. The games
I'm bringing to this are my version of Hyperchess, and the David
Paulowich-mentioned Chess on Two Boards. The comparison of these last 2
games illustrates where I got some of my ideas from. The comparison with
Ben's TessChess, which we are both working on now... well, we're pushing
pieces and I'm making some strange suggestions. 

David P and I have discussed some higher-D ideas, and his 3D games propose
some ideas that promise interesting times in the chesscube - 3D Grand
Acedrex, for example:
http://play.chessvariants.org/pbm/play.php?game%3DGrande+Acedrex+2007%26settings%3DGrande+Acedrex+2007

Check out his rules, in the 3D chess thread. Those games are worth looking
at. Charles Gilman has proposed a radical shortrange 3D game himself, that
is ready to playtest. And those games that aren't quite done? David C,
Hafsteinn, set up a board with some pieces and push the pieces and kick
some ideas around in company. Or we could kick the pieces around and push
ideas at one another. As long as there's no blood; it makes the monitor
all sticky and hard to see. I've got a few irons in the 3D fire myself. As
well as 3D Great Shatranj, Directed Alice III is an entry here. 

There's a lot of ideas around, many contradictory. The nice thing about
variants is that all these ideas can find games to demonstrate their
usefulness, we just have to look. Imagine what kind of a tournament we
could hold with the games that come out of here.