This paper investigates robustness properties of gene regulatory networks with cyclic interconnections. In particular, we treat the heterogeneity of the gene's dynamics using a framework of robustness analysis, and derive necessary and sufficient conditions for robust stability. To this end, we first present a mathematical formulation of the gene regulatory network where dynamics of gene expression has a certain degree of uncertainty. Then, graphical and analytic criteria for robust stability are derived based on a robustness analysis scheme for large-scale multi-agent systems. These conditions are easily applicable to large-scale gene regulatory networks, which involve many components. Finally, the results are interpreted from a biological viewpoint, and they are verified by numerical simulations.