Given a graph G and a set of k colors, assign an arbitrary subset of these colors to each vertex of G. If each vertex to which the empty set is assigned has all k colors in its neighborhood, then the assignment is called a k-rainbow dominating function (kRDF) of G. The minimum sum of numbers of assigned colors over all vertices of G is called the k-rainbow domination number of graph G, denoted by γ rk (G). In this paper, we focus on the study of the k-rainbow domination number of the Cartesian product of cycles, C n 2C m . For k ≥ 8, based on the results of J. Amjadi et al. (2017), γ rk (C n 2C m ) = mn. For (4 ≤ k ≤ 7), we give a proof for the new lower bound of γ r4 (C n 2C 3 ). We construct some novel and recursive kRDFs which are good enough and upon these functions we get sharp upper bounds of γ rk (C n 2C m ). Therefore, we obtain the following results:We also discuss Vizing's conjecture on the k-rainbow domination number of C n 2C m .