A hollow-core microstructured polymer optical fiber was analyzed in the terahertz (THz) region. Spectral analysis of time domain data shows propagation of THz waves in both the hollow-core and the microstructured cladding with a time delay of approximately 20 ps. The frequency range and shift of the transmission bands between different sized waveguides suggested photonic bandgap or resonant guidance. Finite-difference time domain calculations agree relatively well to the experimental transmission results. Propagation losses were estimated to be as low as 0.9 dB/cm.