Confirming a conjecture of Gyarfas, we prove that, for all natural numbers k and r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmiklossy [Discrete Math., 340 (2017), pp. 2053-2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number.