The quantum mechanical treatment of the coupled harmonic oscillators redirives the Dulong-Petit result in the limit of high temperatures. For low temperatures it gives the observed 
-term in the specific heat. It's interesting to note that in the quantum mechanical framework the classical normal modes show up again in a quantized version which is called phonon. Therefore interaction with (the dynamics of) ions is often called interaction with phonons instead.
Furthermore in this way all other mentioned failures of the static lattice model can be remedied. The only necessary corrections for some materials are the anharmonic terms.