Nonreciprocal phonon emission is predicted theoretically from the coherent excitation of two coupled optomechanical cavities arranged along a phoxonic crystal nanobeam. The latter consists of a periodic array of holes and stubs and exhibits simultaneous photonic and phononic bandgaps. It is shown that nonreciprocal phonon emission arises from a combined effect of the spatial symmetry of the cavities and their underlying coupled phononic modes and the temporal phase shift between the excitation sources. This demonstration paves the way for the development of advanced integrated phonon networks and circuits, in which mechanical waves connect different elements in phononic and optomechanical structures.