The purpose of a "background-effect" is to give a feeling of "depth" to whatever is in front of it.
But it's rather difficult to figure out the depth of the chessboard, for several reasons:
- there is no shading but a constant 2d-colour-gradient (which is irritating)
- the chesspattern repeats too often, so you can't tell a direction from the visible edges.
- there is no connection between the board and shape in front of it
So in my opinion it doesn't really work well as a background-effect and it's hard to tell anything that could "fit" in front of it.
Maybe I would try to give the chessboard some structure or make bars out of it, like this

- just with a "something cute looking" color-sheme

That could make a nice scroller, too...
Maybe you can make that shape in front of the board some sort of lens to disconnect the background when in- and outside of the shape.