Polymers have many desirable properties for engineering systems-e.g., low mass density, chemical stability, and high strength-to-mass ratio-but applications of polymers in situations where heat transfer is critical are often limited by low thermal conductivity. Here, we leverage the enormous research and development efforts that have been invested in the production of high-modulus polymer fibers to advance understanding of the mechanisms for thermal transport in this class of materials. Time-domain thermoreflectance (TDTR) enables direct measurements of the axial thermal conductivity of a single polymer fiber over a wide temperature range, 80 < T < 600 K. Relaxation of thermoelastic stress in the Al film transducer has to be taken into account in the analysis of the TDTR data when the laser spot size is small because the radial modulus of the fiber is small. This stress relaxation is controlled by the velocity of the zero-order symmetric Lamb mode of a thin Al plate. We find similarly high thermal conductivities of Λ ≈ 20 W m-1 K-1 in crystalline polyethylene and liquid crystalline poly(p-phenylene benzobisoxazole). For both fiber types, Λ(T) 1/T near room temperature, suggesting an intrinsic limit to the thermal conductivity governed by anharmonicity, not structural disorder. Because of the high degree of elastic anisotropy, longitudinal acoustic phonons with group velocities directed along fiber axis are likely to be the dominate carriers of heat.