The role of Dyakonov surface waves in the transmission through structures composed of birefringent media is theoretically explored. In the case of structures using prisms, unexpected high transmission above the critical angle due to resonant excitation of Dyakonov surface waves is predicted. This transmission is produced only when TE polarized incident wave reaches the interface supporting the surface waves within a narrow interval of angles, for both the angle of incidence and the angle with respect to the optic axis of the birefringent media. As a result, over 90% transmission for a single and isolated peak confined in the two transversal directions, with hybrid TE and TM polarization, can be obtained.