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Calculations are reported for the spin-wave bands and gaps in magnonic crystals formed by patterning one (or both) of the surfaces of ferromagnetic thin films. Specifically, a patterning in terms of a periodic array of rectangular grooves is considered, and the theory employs a microscopic (or Hamiltonian-based) method that incorporates exchange and magnetic dipole–dipole interactions, as well as an external magnetic field. The coupling between different periodic elements of the magnonic crystal involves both exchange and dipolar terms, allowing for control of the spin-wave bands and gaps as the depth and width of the grooves are varied. Numerical examples are given for Permalloy structures, considering the stripe patterning to be on one or two surfaces, and the applicability to Brillouin light scattering is discussed.