About eight years ago it was predicted theoretically that a charged chiral plasma could support the propagation of the so-called chiral magnetic waves, which are driven by the anomalous chiral magnetic and chiral separation effects. This prompted intensive experimental efforts in search of signatures of such waves in relativistic heavy-ion collisions. In fact, several experiments have already reported a tentative detection of the predicted signal, albeit with a significant background contribution. Here, we critically reanalyze the theoretical foundations for the existence of the chiral magnetic waves. We find that the commonly used background-field approximation is not sufficient for treating the waves in hot chiral plasmas in the long-wavelength limit. Indeed, the back-reaction from dynamically induced electromagnetic fields turns the chiral magnetic wave into a diffusive mode. While the situation is slightly better in the strongly-coupled near-critical regime of quark-gluon plasma created in heavy-ion collisions, the chiral magnetic wave is still strongly overdamped due to the effects of electrical conductivity and charge diffusion.