  We calculate the single-particle Green's function of electrons that are
coupled to acoustic phonons by means of higher dimensional bosonization. This
non-perturbative method is {\it{not}} based on the assumption that the
electronic system is a Fermi liquid. For isotropic three-dimensional phonons we
find that the long-range part of the Coulomb interaction cannot destabilize the
Fermi liquid state, although for strong electron-phonon coupling the
quasi-particle residue is small. We also show that Luttinger liquid behavior in
three dimensions can be due to quasi-one-dimensional anisotropy in the
electronic band structure {\it{or in the phonon frequencies}}.
