Is this the pattern?
http://www.kayewood.com/Radiant-Suns...s-DPD-RASU.htm
For a scrappy quilt, what I would do is *heavily* starch all the fabrics before cutting. I consider a heavy starch to be a 1:1 solution of Sta-Flo liquid starch and water. Heavy starching will help prevent stretching and distortion of the fabric while you work.
I would not completely ignore the arrows on the templates, but would rather try to match the arrows with the grain of the fabric piece. You can tell the grain of the fabric by looking at it closely (a magnifying glass helps). You don't need to be exactly on grain, but the closer to on-grain you can get the easier the piecing will be. In other words, I wouldn't worry about mixing lengthwise and crosswise grains; I would just want to align the arrows with one grain or the other.