A new theory is proposed to explain the heating of the solar chromosphere, and possibly the corona, by the dissipation of hydrodynamic compression waves. The basis of the dissipative mechanism, here referred to as ionization pumping, is hysteresis caused by irreversible relaxation of the chromospheric medium to ionization equilibrium following pressure perturbations. In the middle chromosphere, where hydrogen is partially ionized, it is shown that ionization pumping will cause strong dissipation of waves whose periods are 200s or less. This could cause heating of the chromosphere sufficient to compensate for the radiative losses. The mechanism retains a high efficiency for waves of arbitrarily small amplitude and, thus, can be more efficient than shock dissipation for small perturbations in pressure. The formation of shocks therefore is not required for the dissipation of waves whose periods are several minutes or less.