Optimal Dimensions for a Checkerboard-style board?
I don't think 'optimal' is easy to answer without getting into personal preference. There are certainly minimal and maximal sizes. A good example of a very small grid would be Halma (or Go). A good example of a large grid would be human chess. As long as you can make out the pieces and see the whole board. So it depends on the size, complexity, variation and even degrees of freedom in movement of your pieces.

Also squares might not be your best option. Go uses rectangles, so that when you view the board at a normal playing angle it looks like a square grid.