We develop a semiclassical theory of sound absorption in Weyl semimetals in magnetic fields. We focus on the contribution to the absorption that stems from the existence of Berry monopoles in the band structure of such materials or, equivalently, due to the chiral anomaly and chiral magnetic effect. Sound absorption is shown to come primarily from the motion of Weyl nodes in energy space, associated with the propagation of a sound wave. We argue that acoustic magnetochiral dichroism, which occurs when absorption of sound is different for opposite propagation directions and for opposite directions of the magnetic field, can be a definitive probe of band topology. The part of the monopole-related sound magnetoabsorption that is even in the magnetic field is negative in time-reversal Weyl materials. The difference in the sign of the effect as compared to the positive anomaly-related transport magnetoconductance stems from the existence of valley electrochemical imbalances without magnetic field in the sound propagation problem. In centrosymmetric Weyl semimetals with few Weyl nodes, the magnetoabsorption is negative at low frequencies but can change signs with increasing frequency.