Cutting pieces, can't eliminate border

[gmitche]

I am trying to make an image of a puzzle with different colored pieces.  In the example attached, I've cut out 2 white and 2 black pieces and put into layers.  Showing those 4 layers, we see the puzzle pattern/outline.  I don't want that.  I have the puzzle template as an svg and can't get the paths so narrow that they won't show in the final output, but not so narrow (or nonexistent) that the fuzzy select will still work.  
I am a newbie so verbose help would be appreciated.
Thanks.