The \(k\)-rainbow domination is a variant of the classical domination problem in graphs and is defined as follows: Given an undirected graph \(G = (V, E)\) and a set of \(k\) colors numbered \(1, 2, \dots, k\), we assign an arbitrary subset of these colors to each vertex of \(G\). If a vertex is assigned the empty set, then the union of color sets of its neighbors must be \(k\) colors. This assignment is called the \(k\)-rainbow dominating function of \(G\). The minimum sum of numbers of assigned colors over all vertices of \(G\), is called the \(k\)-rainbow domination number of \(G\). In this paper, we present some bounds on the \(3\)-rainbow domination number of circulant networks and grid networks.