Summary: We consider linear waves propagating over periodic topographies of arbitrary amplitude and wave form, generalizing the method in [the authors, J. Fluid Mech. 593, 209–234 (2007; Zbl 1128.76012)]. By a judicious construction of a conformal map from the flow domain to a uniform strip, exact solutions of Floquet type can be developed in the mapped plane. These Floquet solutions, in an essentially analytical form, are analogous to the complete set of flat-bottom propagating and evanescent waves. Therefore they can be used as a basis for the solutions of boundary value problems involving a wavy topography with a constant mean water depth. Various concrete examples are given and quantitative results are discussed. Comparisons with experimental data are made, and qualitative agreement is achieved.