This is the solution finished last night. I appliqued 13 leaves around the edges, breaking up the lines. First I tried it by cutting leaves out of purple paper and putting them in various spots. After I was satisfied with the arrangement, I cut the leaves out of two batik fabrics, one slightly darker than the other, and sewed them in place.
Now when I look at it, my eye now goes to the middle, not the lines. At least now I can hang it on the wall without being totally frustrated! If anyone has any other suggestions, please share them!