Here's the problem. If you want the checkerboard squares to line up perfectly with the 9" blocks, then you need the size of the checkerboard squares to divide 9 evenly. Yet the size of the squares must also divide 2 evenly. The only way to accomplish this is to have squares that are 1/2 inch finished.
There are two other ways to go. One would be to change your sashing to 3" (in which case you could use 3" checkerboard squares) or 1.5" (and go for 1.5" squares). The other is to accept checkerboard squares that do not line up perfectly with your 9" blocks.