This article presents a general formulation of an atomistic Green's function (AGF) method. The atomistic Green's function approach combines atomic-scale fidelity with asymptotic treatment of large-scale (bulk) features, such that the method is particularly well suited to address an emerging class of multiscale transport problems. A detailed mathematical derivation of the phonon transmission function is provided in terms of Green's functions and, using the transmission function, the heat flux integral is written in Landauer form. Within this theoretical framework, the required inputs to calculate heat flux are equilibrium atomic locations and an appropriate interatomic potential. Relevant algorithmic and implementation details are discussed. Several examples including a homogeneous atomic chain and two heterogeneous atomic chains are included to illustrate the applications of this methodology.