You can use the fold and flip method. You start in the middle of your quilt and place the first strip face up. Then place the next strip face down on the first, lined up, and sew through all the layers with your 1/4" seam. Then fold the top stip over and press. Put your next strip face down on it, and do another seam through all the layers again. Fold it over and press. Keep going until you're finished that side.
Then switch the quilt over and start in the middle again and work out to the other side.
When all your vertical strips are on, then just run a few horizontal lines of stitching through the layers and add your binding.
Voila, no sashing.