Is the effect you want to have the occasional changes in fabric in some of the rows?
I ask b/c the strips are what...40 inches long? You want a 40 inch square, why not just start sewing strips together along the length? That's what I would do...sew 2 40 inch long strips together, sew another set of two, etc, then sew those to each other to make foursies, and then together again to make groups with 8 rows...till you have a big enough piece. To me that seams easier then doing the jelly roll race quilt.