A parameter of a mathematical model is structurally identifiable if it can be determined from noiseless experimental data. Here, we examine the identifiability properties of two important classes of linear compartmental models: directed-cycle models and catenary models (models for which the underlying graph is a directed cycle or a bidirected path, respectively). Our main result is a complete characterization of the directed-cycle models for which every parameter is (generically locally) identifiable. Additionally, for catenary models, we give a formula for their input-output equations. Such equations are used to analyze identifiability, so we expect our formula to support future analyses into the identifiability of catenary models. Our proofs rely on prior results on input-output equations, and we also use techniques from linear algebra and graph theory.