This paper is devoted to the generic parameters identifiability analysis for structured linear systems with unknown inputs. The proposed method is based on a graph-theoretic approach and assumes only the knowledge of the system's structure. Necessary and sufficient conditions are expressed in graphical terms which ensure that a given set of unknown parameters characterising the model of the system are structurally identifiable using the available measurements, assuming that the inputs and initial conditions are quite informative. Even if we are not interested in proposing identification schemes, these given conditions are necessary and sufficient to the parameter identification whatever the used algorithms to determine numerically the parameters values. Finally, note that the given conditions are quite easy to check because they are based on searching and comparing some specific paths and cycles in a digraph. This makes our approach well suited to study large-scale systems.