We derive analytic expressions for the classical Casimir-Polder free energy and force for a polarizable (magnetizable) atom (microparticle) interacting with thin films, made of different materials, or graphene. It is shown that for an isolated dielectric film the free energy and force decrease quicker with separation, as compared to the case of atom interacting with a thick plate (semispace). For metallic films some peculiar features depending on the model of a metal used are analyzed. For an atom interacting with graphene we demonstrate that at room temperature the classical regime is achieved at about $1.5\,\mu$m separation. In this regime the contributions to the free energy and force due to atomic magnetic polarizability are suppressed, as compared to main terms caused by the atomic electric polarizability. According to our results, at separations above $5\,\mu$m the Casimir-Polder interaction of atoms with graphene is of the same strength as with an ideal-metal plane. The classical interaction of atoms with thin films deposited on substrates is also considered.