I clip seams after sewing each row of squares. Then I clip each row seams when the rows are sewn together. Then when I am done, all I have to do is clip around the entire quilt. And, as others suggested, don't try to do it all at once. Sew a row, clip, rest. Sew another row, clip, rest. Sew these 2 rows together, clip, rest. I think you get the picture.
A more expensive option, but definitely worth it if you plan to make many rag quilts is to get the Accucut GO! They have a die that is perfect for rag quilts.