Sisal ropes are inexpensive ($4 for 500 ft), strong and natural (made of plant fiber).
They are available at hardware stores (Home Depot, Lowe's, Menards) and Amazon.
I use them for all sorts of trellising and the Florida weave.
Blue Hawk 525-ft Twisted Sisal Rope at Lowes.com http://low.es/13Ectxr
Crown Bolt #21 x 525 ft. Natural Sisal Bundling Twine-65525 at The Home Depot http://thd.co/13Ecqlh