It has been sometimes necessary for personnel to work in areas where low-energy heavy ions interact with targets or with beam transport equipment and thereby produce significant levels of radiation. Methods to predict doses and to assist shielding design are desirable. The Particle and Heavy Ion Transport code System (PHITS) has been typically used to predict radiation levels around high-energy (above 100 MeV amu(-1)) heavy ion accelerator facilities. However, predictions by PHITS of radiation levels around low-energy (around 10 MeV amu(-1)) heavy ion facilities to our knowledge have not yet been investigated. The influence of the "switching time" in PHITS calculations of low-energy heavy ion reactions, defined as the time when the JAERI Quantum Molecular Dynamics model (JQMD) calculation stops and the Generalized Evaporation Model (GEM) calculation begins, was studied using neutron energy spectra from 6.25 MeV amu(-1) and 10 MeV amu(-1) (12)C ions and 10 MeV amu(-1) (16)O ions incident on a copper target. Using a value of 100 fm c(-1) for the switching time, calculated neutron energy spectra obtained agree well with the experimental data. PHITS was then used with the switching time of 100 fm c(-1) to simulate an experimental study by Ohnesorge et al. by calculating neutron dose equivalent rates produced by 3 MeV amu(-1) to 16 MeV amu(-1) (12)C, (14)N, (16)O, and (20)Ne beams incident on iron, nickel and copper targets. The calculated neutron dose equivalent rates agree very well with the data and follow a general pattern which appears to be insensitive to the heavy ion species but is sensitive to the target material.