A new simple method for computing the attenuation (due to dielectric losses) of weakly damped electromagnetic waves in a wide class of wave-guiding structures is suggested. The wave guiding structures considered contain time dispersive dielectric media, and restrictions are set neither on the type of modes nor on the cross-section geometry. Only one of the dielectrics of the structure is supposed to be lossy. The wave attenuation coefficient results directly from the slope of the wave dispersion presented as a dependence of the wave number on the permittivity of this dielectric at fixed wave frequency. The method is demonstrated on structures containing cold isotropic plasmas. For this case the method was found to be applicable even for some configurations with inhomogeneous plasma filling.