@Bn Em, I think that I'm late to reply to this, but thank you for clarifying that.
John J. G. Savard, who runs quadibloc.com stated that pieces on triangular tiling 'naturally go around in circles' if direct 'hexagonal' logic will be applied as you described it; he also suggested that alternating movement can be seen as a solution, but no diagrams of it were provided although I believe that this idea can be executed similarly to mine.
A one-direction movement which can be seen as an alternative to the alternating method doesn't work equally for both the rook and bishop. So, I think that my goal on this is completed.
I didn't know what to do with knight and pawn, so I simply followed the logic of orthodox chess.
@Bn Em, I think that I'm late to reply to this, but thank you for clarifying that.
John J. G. Savard, who runs quadibloc.com stated that pieces on triangular tiling 'naturally go around in circles' if direct 'hexagonal' logic will be applied as you described it; he also suggested that alternating movement can be seen as a solution, but no diagrams of it were provided although I believe that this idea can be executed similarly to mine. A one-direction movement which can be seen as an alternative to the alternating method doesn't work equally for both the rook and bishop. So, I think that my goal on this is completed.
I didn't know what to do with knight and pawn, so I simply followed the logic of orthodox chess.