This paper introduces a multiplicity-preserving triangular set decomposition algorithm for systems of three polynomials. For unmixed and regular algebraic varieties defined by three polynomials, the algorithm decomposes them into a sum of square-free and disjoint algebraic cycles represented by triangular sets. A novel aspect of this approach is that, in this setting, the algorithm primarily relies on the computation of primitive polynomial remainder sequences instead of Gröbner bases. For the case of three variables, techniques such as sum-and-quotient decomposition of ideals and other alternative decomposition methods are employed to handle irregular cases. A complete algorithm is presented for decomposing the system into zeros represented by triangular sets with multiplicities. Relevant experimental examples are provided to illustrate the efficiency of the proposed algorithm.