Abstract: We build a class of invisible axion models with tree-level Flavor Changing Neutral Currents completely controlled by the fermion mixing matrices. The scalar sector of these models contains three-Higgs doublets and a complex scalar gauge singlet, with the same fermionic content as in the Standard Model. A horizontal Peccei-Quinn symmetry provides a solution to the strong CP problem and predicts the existence of a very light and weakly coupled pseudo-Goldstone boson, the invisible axion or familon. A phenomenological analysis is performed taking into account familon searches in rare kaon and muon decays, astrophysical considerations and axion searches via axion-photon conversion. Drastic differences are found in the axion properties of different models due to the strong hierarchy of the CKM matrix, making some of the models considered much more constrained than others. We also obtain that a rich variety of these models avoid the domain wall problem. A possible mechanism to protect the solution to the strong CP problem against gravitational effects is also discussed.