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Even in 3D chess, we approach the limits of human observance, and the
ability to form deep strategies.

If ever we are to play more interesting (or at least more complex) forms of
chess, we have to think of another way of playing it, not in terms of
individual moves, which become too cumbersome, but in terms of commands,
and layered on top of that, wider strategies for games of ever increasing
intricacy.

Working at the core of this would be a running index for each piece-a heads
up display. For example for a starting rook in 6D chess you could have:

************

111111 (coordinate index)

100000 (move type index/previous move made)

- (opponent piece type threatened by-ABC etc if applicable)

abcdef (friendly piece types in range of rooks movement)

- (other special piece status eg P=pinned)

************

A FIDE array would read: 
11/10/-/N,p1;12/1:2/-/p3;13/11/-/p, p; 14/10, 11/-/B, K, p, p ,p; etc

Okay, not terribly pretty, but the basis for a much more streamlined way to
play an advanced game. Simple macrocommands could include; skirmish (move
to furthest unprotected enemy square), kick (threaten material deficit),
spread out (increase mobility of side), retreat, block, pin, retaliate,
check, double up etc.

The next layer would be how to bundle all that into higher strategies.

Note: I'm not much of a programmer, but maybe this could form the basis of
an efficient and flexible AI?