Strip piecing usually means one of two ways:
Sew the strips on to paper, interfacing, wash away stabilizer, or the new stay in stabilizer. Remove the paper and use the strip block as any other block.
or
Put batting on top of the backing fabric and sew your strips together on top of the batting and backing. When you get your block sewn it's completely quilted.
I have never paid any attention to grain when strip piecing.
I think the fencing you are referring to is what most call sashing? I'd love to see a picture of your blocks.