I think the other posts have mentioned most of the natural fibers used for rope. When you are talking about ropes of 1/4 inch diameter or less, you are probably going to marline is the easiest to find. In larger size, manilla would probably become easier to find. If you cannot find it through a marine supplier, try to contact a ship's chandler. If that failed, try calling a maritime academy and ask someone there who teaches marlinspike seamanship about where they obtain natural fiber lines (maybe try http://www.imq.qc.ca/eng//)

All that said, I cannot see using natural fiber for anythign that I really need to count on. Any line that you buy should be inspected and renewed from time to time, but natural fibers of there very nature wear out much faster than synthetics, both when in use and not in use. They rot. A natural fiber rope can become useless without ever having been used or mistreated. So, in addition to not being as strong or useful, natural fibers wear out faster.