We have investigated the effects of spatial-inversion and time-reversal symmetry breaking on the acoustic phonon branches in chiral MnSi using high-resolution inelastic x-ray scattering and first-principles calculations. We find a momentum-transfer-dependent (q-dependent) splitting between transverse phonon bands having angular momentum parallel or antiparallel to q. This is understood by a phenomenological theory using a gyrotropic tensor. We observed no significant impact from the time-reversal symmetry breaking induced by a magnetic field (energy shifts < 0.3 meV). This suggests the effect of time-reversal symmetry breaking is small or is restricted to a very-low-energy regime in this material, possibly due to small spin-orbit interaction.