We develop a method to prove the existence of multisite localized periodic oscillations in networks of weakly coupled autonomous oscillators. First, we obtain a necessary condition and a sufficient condition for continuation of a degenerate periodic orbit. These two conditions are applied to the case of m-site periodic oscillations in the uncoupled limit, which are all degenerate and form an m-torus parametrized by phases of oscillations. The result is similar to Melnikov theory, i.e. some periodic orbits in the torus are continued for weak coupling when their relative phases are associated with zeros of two vector valued functions calculated from the solutions in the uncoupled limit. The continued periodic orbits are spatially exponentially localized for exponentially decaying coupling. This method is applied to both conservative and nonconservative oscillator networks. In particular, if oscillations in local dynamics are described by an action-angle form in Hamiltonian networks, the relative phase of the continued periodic orbit is associated with a nondegenerate critical point of a scalar function related to the effective action.