We construct an infinite family of closed 3-manifolds M(2m+ 1, n, k) which are identification spaces of certain polyhedra P(2m+ 1, n, k), for integers $m;ge;1,;n;ge;3,;and;k;ge;2$. We prove that they are (n / d)- fold cyclic coverings of the 3-sphere branched over certain links $L_{(m,d)}$, where d = gcd(n, k), by handle decomposition of orbifolds. This generalizes the results in [3] and [2] as a particular case m = 2.