Based on a method for right coprime factorizations of linear systems using matrix elementary transformations, it is shown that a very simple iteration formula exists for right coprime factorizations of multi-input linear systems in system upper Hessenberg forms. This formula gives directly the coefficient matrices of the pair of solutions to the right coprime factorization of the system Hessenberg form, and involves only manipulations of inverses of a few triangular matrices and some matrix productions and summations. Based on this formula, a simple, efficient procedure for determining a right coprime factorization of a multi-input linear system is proposed, which first converts a given linear system into its system Hessenberg form using some orthogonal similarity transformations and then applies the iteration formula to the converted system Hessenberg form. An example demonstrates the usage of the approach.