The Physics of a Chess Board

In Through the Looking Glass by the Reverend Lewis Carol, Alice walks through a mirror in her living room and finds the chessboard that normally resides there to be teeming with little chess pieces running around.  Leaving her mirror-house, the entire country around it has been transformed in to a chessboard.  Alice starts as a pawn and has to walk forward one step at a time to the end, when she will become a queen and be able to run as fast as she wants across the country.

While Carol's story is whimsical and fun, what would be the implications of living in a chess board?  What are the "physical laws" experienced by a given chess piece?

So imagine all the universe to be a discrete 8X8 grid, alternately tiled with black and white, and conceive of a chess piece as being a kind of elementary particle in this bizarre chess world.  We will look mostly at the free dynamics of such a chess particle - that is, how it behaves dynamically in the absence of other pieces.