The absorption of electromagnetic waves by a thin dielectric slab backed by a metasurface has been comprehensively studied and discussed at microwave frequencies. The metasurface consists of a metallic plate decorated with a periodic distribution of coaxial- or annular-type cavities. Analytical expressions for the absorbance have been obtained by using a mode-matching method. For P-polarized waves it is predicted that a low-frequency peak of perfect absorption is possible by properly choosing the lossy component of the permittivity of the dielectric slab, with the position of this peak being easily tunable by the cavity length. It is also shown that non-Bravais lattices, containing several cavities in the unit cell, and the excitation of guided waves by the periodic array of resonators are complementary absorption mechanisms in the studied structures.