Obtaining the complete weight distributions for nonbinary codes is an even harder problem than obtaining their Hamming weight distributions. In fact, obtaining these distributions is a problem that usually involves the evaluation of sophisticated exponential sums, which leaves this problem open for most of the linear codes. In this work we present a method that uses the known complete weight distribution of a given cyclic code, to determine the complete weight distributions of other cyclic codes. In addition we also obtain the complete weight distributions for a particular kind of one- and two-weight irreducible cyclic codes, and use these distributions and the method, in order to determine the complete weight distributions of infinite families of cyclic codes. As an example, and as a particular instance of our results, we determine in a simple way the complete weight distribution for one of the two families of reducible cyclic codes studied by Bae, Li and Yue [Discrete Mathematics, 338 (2015) 2275-2287].