There's a good chance that linen in an epoxy matrix would be a decent composite.
Bamboo fiber, for instance, can be made into a quality composite, even if it isn't optimal for a bike frame.
Fiber in a polymer matrix is an extraordinary technology. Carbon fiber is an extraordinary fiber, but there are many other fibers to choose from. And indeed, we may have to choose new fibers as our Perpetual War Strategy™ seems to be
using up all the CF available.
™"Perpetual War" is a trademark of the Mao corporation, used with permission by the Bush administration.