Context. The origin of the anomalous microwave emission is not yet identified but it may be associated with (spinning) dust. Aims. We point out that the emission from low-energy, solid-state structural transitions, apparent over a range of dust temperatures, and particularly important at low dust temperatures (T d < 30 K), could provide an origin for the excess microwave emission. Methods. The physics of two-level systems (TLS) within amorphous materials is applied within the context of interstellar dust. Results. TLS systems, which are thought to be at the origin of the temperature-dependent emissivity of interstellar dust, should reveal their existence at long wavelengths in the emission from dust. Conclusions. Low-energy, structural transitions in amorphous interstellar grains could make a contribution to the anomalous microwave emission, which should correlate with the emission from dust components with temperatures less than about 30 K.