The development of the Monte Carlo code SHIELD-HIT (heavy ion transport) for the simulation of the transport of protons and heavier ions in tissue-like media is described. The code SHIELD-HIT, a spin-off of SHIELD (available as RSICC CCC-667), extends the transport of hadron cascades from standard targets to that of ions in arbitrary tissue-like materials, taking into account ionization energy-loss straggling and multiple Coulomb scattering effects. The consistency of the results obtained with SHIELD-HIT has been verified against experimental data and other existing Monte Carlo codes (PTRAN, PETRA), as well as with deterministic models for ion transport, comparing depth distributions of energy deposition by protons, 12C and 20Ne ions impinging on water. The SHIELD-HIT code yields distributions consistent with a proper treatment of nuclear inelastic collisions. Energy depositions up to and well beyond the Bragg peak due to nuclear fragmentations are well predicted. Satisfactory agreement is also found with experimental determinations of the number of fragments of a given type, as a function of depth in water, produced by 12C and 14N ions of 670 MeV u(-1), although less favourable agreement is observed for heavier projectiles such as 16O ions of the same energy. The calculated neutron spectra differential in energy and angle produced in a mimic of a Martian rock by irradiation with 12C ions of 290 MeV u(-1) also shows good agreement with experimental data. It is concluded that a careful analysis of stopping power data for different tissues is necessary for radiation therapy applications, since an incorrect estimation of the position of the Bragg peak might lead to a significant deviation from the prescribed dose in small target volumes. The results presented in this study indicate the usefulness of the SHIELD-HIT code for Monte Carlo simulations in the field of light ion radiation therapy.