I've just dealt with this issue in one quilt. The quilt from he!! btw! My border fabric was very directional. In order to have the borders work correctly I needed to do both lengthwise and crosswise grains/cuts. To be honest, I did not notice much difference in the stretch. This is LQS fabric...don't remember manufacturer. In fact, I tore the lengthwise pieces and have the least amount of 'waffling' on the edges of any other fabric I've done this with previously.
And now tomorrow I will spend close to 2 hrs. on the road to pick up more backing fabric, as the pattern is about 1 yard off on that requirement!