The dynamic nonlinear problem of wave propagation in infinitely long hysteretic beams under bending is addressed. Such a problem is tackled by proposing an asymptotic treatment based on the method of multiple scales. The equations of motion are obtained reducing the three-dimensional-continuum model to the one-dimensional plane beam problem via the cross-section rigidity constraint, and describing the hysteresis via a nonlinear viscoelastic material model, which can be easily tuned to obtain either hardening or softening characteristic responses. Geometric nonlinearities are not taken into account. In addition to the perturbation treatment of the wave equations with hysteretic nonlinearity, which provided the slow amplitude and phase modulations with time and space, a number of numerical tests are presented to show how the hysteretic nonlinear response can give rise to dissipative nonlinear bending waves. The limit case of zero dissipation is also investigated to shed light on the distinct effects of viscolelasticity.