The thermal conductivity of wurtzite and zinc blende indium arsenide nanowires was measured using a microfabricated device, with the crystal structure of each sample controlled during growth and determined by transmission electron microscopy. Nanowires of both phases showed a reduction of thermal conductivity by a factor of 2 or more compared to values reported for zinc blende indium arsenide bulk crystals within the measured temperature range. Theoretical models were developed to analyze the measurement results and determine the effect of phase on phonon transport. Branch-specific phonon dispersion data within the discretized first Brillouin zone were calculated from first principles and used in numerical models of volumetric heat capacity and thermal conductivity. The combined results of the experimental and theoretical studies suggest that wurtzite indium arsenide possesses similar volumetric heat capacity, weighted average group velocity, weighted average phonon-phonon scattering mean free path, and anharmonic scattering-limited thermal conductivity as the zinc blende phase. Hence, we attribute the differing thermal conductivity values observed in the indium arsenide nanowires of different phases to differences in the surface scattering mean free paths between the nanowire samples.