A study of the onset of superconducting phase coherence in periodic wire networks with weakly coupled adjacent nodes is presented. In a narrow temperature range below the mean-field critical temperature, ac conductance measurements reveal a drop in superfluid density and a peak in dissipation providing the first observation in networks of the vortex unbinding transition predicted by the Kosterlitz-Thouless theory. The ac response in the critical region is accurately described by dynamical extensions of the theory incorporating vortex-pinning phenomena created by the periodic structure of the network.