Chalcogenide semiconductors and semimetals are a fertile class of efficient thermoelectric materials, which, in most cases, exhibit very low lattice thermal conductivity κph despite lacking a complex crystal structure such as the tetragonal binary compound InTe. Our measurements of κph(T) in single-crystalline InTe along the c axis show that κph exhibits a smooth temperature dependence upon cooling to about 50 K, the temperature below which a strong rise typical for dielectric compounds is observed. Using a combination of first-principles calculations, inelastic neutron scattering (INS), and low-temperature specific heat and transport properties measurements on single-crystalline InTe, we show that the phonon spectrum exhibits well-defined acoustic modes, the energy dispersions of which are constrained to low energies due to distributions of dispersionless, optical modes, which are responsible for a broad double peak structure in the low-temperature specific heat. The latter are assigned to the dynamics of In+ cations in tunnels formed by edge-sharing (In3+Te42−)− tetrahedra chains, the atomic thermal displacement parameters of which, probed as a function of temperature by means of single-crystal x-ray diffraction, suggest the existence of a complex energy potential. Indeed, the In+-weighted optical modes are not observed by INS, which is ascribed to the anharmonic broadening of their energy profiles. While the low κph value of 1.2Wm−1K−1 at 300 K originates from the limited energy range available for acoustic phonons, we show that the underlying mechanism is specific to InTe and argue that it is likely related to the presence of local disorder induced by the In+ site occupancy.