In this paper, we derive an analytic criterion for the existence of periodic oscillations in cyclic gene regulatory networks where each protein represses another gene expression in a cyclic way. We analyze local instability properties of large-scale cyclic gene regulatory networks based on the result which shows that local instability of an equilibrium point globally guarantees the existence of periodic oscillations. The developed criterion has the following distinctive features: (i) it is applicable to cyclic gene regulatory networks consisting of any number of genes, (ii) the dependence of an equilibrium state on biochemical parameters is explicitly considered, and (iii) it depends only on the given biochemical parameters. Thus, the derived condition can explicitly show how each parameter affects the existence of periodic oscillations. Furthermore, it provides us with novel biological insight on decisive physical quantities for the existence of periodic oscillations, and their quantitative relations with periodic oscillations.