  A system of three quantum particles on the three-dimensional lattice $\Z^3$
with arbitrary "dispersion functions" having non-compact support and
interacting via short-range pair potentials is considered. The energy operators
of the systems of the two-and three-particles on the lattice $\Z^3$ in the
coordinate and momentum representations are described as bounded self-adjoint
operators on the corresponding Hilbert spaces. For all sufficiently small
nonzero values of the two-particle quasi-momentum $k\in (-\pi,\pi]^3$ the
finiteness of the number of eigenvalues of the two-particle discrete
Schr\"odinger operator $h_\alpha(k)$ below the continuous spectrum is
established. A location of the essential spectrum of the three-particle
discrete Schr\"odinger operator $H(K),K\in (-\pi,\pi]^3$ the three-particle
quasi-momentum, by means of the spectrum of $h_\alpha(k)$ is described. It is
established that the essential spectrum of $H(K), K\in (-\pi,\pi]^3$ consists
of a finitely many bounded closed intervals.
