I use a thin really cheap muslin as a foundation for a several of reasons: 1: I just don't want to spend time tearing paper off, 2: I want the ability to make a quilt without batting if I choose, and 3: I also think it stabilizes the block. I just cut my fabric into strips without regard to grain and think those stretchy strips without foundation could get messy to work with in joining the blocks. All this is just my preference, that is the beauty of string quilts, you can do what you want to and it will generally work out.
If you go on Pinterest, do a search for string quilts and be amazed at all the different patterns that can be achieved with just strings. Also a Google search will get you a lot of ideas and tutorials on the many ways to make a string quilt.