This paper provides a detailed study of transmission zeros in the context of linear structured systems. For such systems, the entries of the state space model matrices are supposed to be either fixed zeros or free parameters. A directed graph can be associated to such systems and structural properties can be studied, i.e. properties which are valid for almost all values of the parameters. We show that there exists two kinds of transmission zeros: the parameter dependent ones and the constant ones which can only occur in zero. Moreover, the constant transmission zeros can be directly obtained on the associated graph in terms of particular paths and cycles.