We show that a hot rotating fluid of relativistic chiral fermions possesses a new gapless collective mode associated with coherent propagation of energy density and chiral density waves along the axis of rotation. This mode, which we call the Chiral Heat Wave, emerges due to a mixed gauge-gravitational anomaly. At finite density the Chiral Heat Wave couples to the Chiral Vortical Wave while in the presence of an external magnetic field it mixes with the Chiral Magnetic Wave. The coupling of the Chiral Magnetic and Chiral Vortical Waves is also demonstrated. We find that the coupled waves — which are coherent fluctuations of the vector, axial and energy currents — have generally different velocities compared to the velocities of the individual waves.