We have shown theoretically that a combination of cross-section modulation and acoustic mismatch in the core-shell Si/Ge nanowires can lead to a drastic reduction of the thermal conductivity. Our calculations, which utilized two different models-five-parameter Born-von Karman and six-parameter valence force field - for the lattice vibrations, indicate that the room temperature thermal conductivity of Si/Ge cross-section modulated nanowires is almost three orders of magnitude lower than that of bulk Si. Thermal flux in the modulated nanowires is suppressed by an order of magnitude in comparison with generic Si nanowires. The effect is explained by modification of the phonon spectra in modulated nanowires leading to decrease of the phonon group velocities and localization of certain phonon modes in narrow or wide nanowire segments. The thermal conductivity inhibition is achieved in nanowires without additional surface roughness and, thus, potentially reducing degradation of the electron transport. Our results suggest that the acoustically mismatched cross-section modulated nanowires are promising candidates for thermoelectric applications.