Nanowire diodes have been realized by employing an axial heterojunction between InAs and InSb semiconductor materials. The broken-gap band alignment (type III) leads to a strong rectification effect when the current-voltage (I-V) characteristic is inspected at room temperature. The additional insertion of a narrow InP barrier reduces the thermionic contribution, which results in a net decrease of leakage current in the reverse bias with a corresponding enhanced rectification in terms of asymmetry in the I-V characteristics. The investigated diodes compare favorably with the ones realized with p-n heterostructured nanowires, making InAs/InP/InSb devices appealing candidates to be used as building blocks for nanowire-based ultrafast electronics and for the realization of photodetectors in the THz spectral range.