Radiation from acoustic sources located inside randomly layered structures is studied using the transfer matrix method. It is shown that in contrast to the periodically layered cases where the radiation can be either enhanced or inhibited depending on the frequency and the characteristics and the material composition of the structures, in the random structures the radiation is always inhibited. The degree of inhibition depends on the acoustic frequency, number of random layers, and the randomness and acoustic parameters of the structures. Both spherically and cylindrically random structures are considered. The results point to the possibility of designing sonic waveguide devices that will not suffer from the energy loss caused by radiation, thus allowing effective energy confinement or long-range energy propagation.