We give a new method in order to obtain Weil-Serre type hounds on the minimum distance of arbitrary cyclic codes over F(pe) of length coprime to p, where e >= 1 is an arbitrary integer. In an earlier paper we obtained Weil-Serre type bounds for such codes only when e = 1 or e = 2 using lengthy explicit factorizations, which seems hopeless to generalize. The new method avoids such explicit factorizations and it produces an effective alternative. Using our method we obtain Weil-Serre type bounds in various cases. By examples we show that our bounds perform very well against Bose-Chaudhuri-Hocquenghem (BCH) bound and they yield the exact minimum distance in some cases.