I have done string piecing without any foundation....but only when the strings were fairly wide and mostly on grain. you don't want edges off grain because they will stretch. I used my cutting mat as a guide on how big to make the string squares, made them oversized and then cut down with my ruler. I used a wider string at the top and the bottom so I had enough to square it up without cutting into seams.
I have used muslin ...but that really made the finished quilt heavy and you would have to wash and press first. I did not care for this method.
and I have made string blocks with phone book pages. I actually like this method best. I have not had problems with any ink rubbing off like you would get with newspaper....the older the phonebook, the better I hear as the ink is well set with age. I use smaller stitches so the paper is easy to tear off and do the tearing process while watching TV...it goes pretty quickly. I will also use a crochet hock or chopstick or something to get at the paper behind those narrow strings...I find starting at one edge and then working across the square works well as you have one lose edge of paper as you move across... Some folks think that phone book paper is the easiest to remove...and it has been the best for me.