Thanks to all for your help. I did try bigger triangles but that didn't seem to do the trick. I came across this tute by Bonnie at Quiltville:
http://quiltville.com/sqinsqtutorial.shtml. She has a chart for different sizes, and her inner square for a 5" finished block was 4", whereas the pattern I am using called for a 4 1/8 inner square. Bonnie also said to use bigger triangles and use the Easy Angle. Anyway I tried both size inner squares and bigger triangles and couldn't get it to come out with a 1/4" seam, so that is why I think paper piecing may do the trick.