A nematic liquid-crystal cylinder may exhibit metastable cholesteric-type twisted states in which the director uniformly rotates along the sample in the plane orthogonal to the cylinder axis. If a constant torque is maintained by strongly anchoring end plates, thermal fluctuations can produce rotation, leading to torsionally viscous behavior. We present a theory for the fluctuation-induced intrinsic torsional viscosity of narrow cylinders of nematic, based on the Frank free energy and constructed by analogy with the work of Langer and Ambegaokar on the resistance of superconducting channels.