We propose a scheme to generate coherent radiation with a tunable frequency spectrum based on the excitation of an inhomogeneous thin layer of a resonant medium by an ultrashort pulse. The Cherenkov-type radiation emitted in a transient stage, i.e., while the excitation wavefront is still propagating over the medium, is shown to contain additional frequency components, which are determined by the spatial frequency components of the medium's density distribution and the excitation geometry. We demonstrate that the spectral content of this transient radiation can be adjusted in wide limits by controlling the spatial density distribution of the resonant medium and the shape of the excitation wavefront. Our theoretical results pave the way towards simple and compact swept-frequency or frequency-tunable optical oscillators as well as sources of the frequency-modulated optical radiation.