I would just straight-line quilt it, perhaps on the diagonal if your center white strips are making a lattice pattern on the quilt. You could just make diagonal lines in one direction or in a crisscross pattern.

I quilted one I made with circles, I like the way it turned out. It caught all the seams but not the centers so it went pretty easy.

My two were QAYG with small sashes in between. So they were already quilted.
On other strip quilts I just do an all over large stipple/meander. The seams aren't any thicker than any other pieced quilt. They turned out very well.

I usually tie my string quilts. I have never hand quilted one because of all the seams. On most quilts hand quilting is my chosen method.

When I do my string quilts I cut the batting to the size I want the square and sew my strings directly onto it and therefore I do not need to quilt it as it is already quilted. I just add my backing and binding, the backing has never shifted but these are not large bed size quilts that I do this way.

I have made several string block quilts. I used the "quilt as you go method". Your strips are usually not over 2" and quilting really isn't necessary. I did want to anchor the strips connecting the blocks and just stitched a curve down the middle of each. Real easy and real quick. You can arrange the blocks to make patterns, piecing light strips on half and darker strips on the other half.

quilting string quilts
 

Check out http://heartstringsquiltproject.com/ We make string quilts for charity and have loads of examples on the photo pages.