The strip idea will only work for the center blocks. Sew 2 strips of one color to each side of a second color then sub cut into the second row. You're still going to have to make all the half square triangles (HST's) and put a color between them.
For 3 fabrics, the easiest blocks I can think of to sew in strips would be 1. Rail Fence, 2. 9 Patch or Disappearing 9 Patch, 3. Warm Wishes, 4. "H" block. 5. Tube Blocks which are cool and look a lot harder than they are--use starch, though so the bias edges don't go wonky on you.