This Rapid Communication addresses the relationship between hyperbolic equations of heat conduction and microscopic models of dielectrics. Effects of the nonstationary heat conduction are investigated in two one-dimensional models with conserved momentum: Fermi-Pasta-Ulam (FPU) chain and chain of rotators (CR). These models belong to different universality classes with respect to stationary heat conduction. Direct numeric simulations reveal in both models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with parabolic Fourier equation of the heat conduction. The crossover wavelength decreases with increase in average temperature in both models. For the FPU model the lowest-order hyperbolic Cattaneo-Vernotte equation for the nonstationary heat conduction is not applicable, since no unique relaxation time can be determined.