DAN+, Different Kings and Pawns?Changing the move of the King or the Pawns is dangerous.
For example, if one side has a normal King, and the other side has a King that jumps one or two squares Rookwise, the side with the normal King will win every King-and-Pawn endgame.
Another example is Berolina Chess, in which Pawns move diagonally but capture Rookwise (just the reverse of normal Pawns); the opening move for White of 1. a2-c4 or 1. h2-f4 is strongly indicated if White has normal Rooks!
Changing the Pawns would prevent the creation of locked Pawn chains, which is important to the look and feel of Chess.
All in all, I cannot currently recommend substitutes for the standard King and pawns.