Data on neutron dose attenuation by thick concrete, cast iron, and cast iron plus concrete composite shields for heavy ions and protons having high energies (200–1000 MeV/u) are necessary for shielding designs of high-powered heavy ion accelerator facilities. Neutron production source terms, shield material attenuation lengths, and neutron dose rate reduction effectiveness of the bulk shielding in the angular range from 0° to 125° were determined by the Particle and Heavy Ion Transport Code (PHITS) for beams of 300 and 550 MeV/u 48Ca ions, 200 and 400 MeV/u 238U ions, 800 MeV/u 3He and 1 GeV protons. Calculated results of interaction lengths of concrete and cast iron were also compared with similar work performed by Agosteo et al., and to experimental and other calculated data on interaction lengths. The agreement can be regarded as acceptable.