In recent years there has been growing interest in verifying the horizon-scale homogeneity of the Universe that follows from applying the Copernican Principle to the observed isotropy. This program has been stimulated by the discovery that a very large void, centred near us, can explain supernova luminosity distance measurements without dark energy. It is crucial to confront such models with as wide a variety of data as possible. With this application in mind, I develop the relativistic theory of linear scalar perturbations on spherically symmetric dust (Lemaitre-Tolman-Bondi) spacetimes, using the covariant 1 + 1 + 2 formalism. I show that the evolution of perturbations is determined by a small set of new linear transfer functions. If decaying modes are ignored (to be consistent with the standard inflationary paradigm), the standard techniques of perturbation theory on homogeneous backgrounds, such as harmonic expansion, can be applied, and results closely paralleling those of familiar cosmological perturbation theory can be obtained.

16 pages, 3 figures; numerical results and discussion expanded, one figure added; version accepted to Phys. Rev. D