When I cut out the pieces, I evenly divide each color or fabric into 4 stacks (quadrants). Then I pick up and sew randomly. I use all the fabrics in each quadrant before moving on to the next making sure they are evenly distributed. After getting the rows sewed, by quadrant, I lay them out. if I have to move something, I just flip a row. I hope this makes sense.
I have used the paper plate thing, when I do have to lay it out, and i have to say, it works really well. I draw an arrow on the plate to show which direction to iron each row so they nest well. Don't forget to number the paper plate with the row number. Saves a lot of confusion. I re-use them, too.