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Few CVs have triangles. They work symmetrically for two-player  .
Chess as well as three. Here is suggested starting array   1-> ._. 
for what is called to right 8^8^8 in 64 cells.   234->        ._._. 
K50,64. Q51,63. R37,38,48,49.                  56789->       ._._._.    
The first are Black and second half White.  10 to 16->      ._._._._.
B26,27,35,36. N17,18,24,25.                 17 to 25->     ._._._._._.
Pawns in front 1 or 2 step, then 1 only.    26 to 36->    ._._._._._._.
Promotion zone Black is triangles           37 to 49->   ._._._._._._._.
4,9,16,25,36,49,64, and White across        50 to 64->  ._._._._._._._._.
correspondingly. All F.i.d.e. rules apply. Notice maximum Rook move is same 7 cells as in squares. Rook's path from corner 50 is 50-51-38-27-18-11-6-3. In these equilateral triangles, the return path is different, 3-2-5-10-17-26-37-50, with all the implications.  Jonathan was on the right track at Spratt's Chess for Three: 
http://www.chessvariants.org/index/displaycomment.php?commentid=18123.