We derive a somewhat crude, yet very efficient semiclassical approximation for computing nonadiabatic spectra. The resulting method, which is a generalization of the multiple-surface dephasing representation, includes quantum effects through interference of mixed quantum-classical trajectories and through quantum treatment of the collective electronic degree of freedom. The method requires very little computational effort beyond the fewest-switches surface hopping or Ehrenfest locally mean-field dynamics and is very easy to implement. The proposed approximation is tested by computing the absorption and time-resolved stimulated emission spectra of pyrazine using the four-dimensional three-surface model which allows for comparison with the numerically exact quantum spectra. As expected, the multiple-surface dephasing representation is not suitable for high-resolution linear spectra, yet it seems to capture all the important features of pump-probe spectra. Finally, the method is combined with on-the-fly ab initio evaluation of the electronic structure (i.e., energies, forces, electric-dipole, and nonadiabatic couplings) in order to compute fully dimensional nonadiabatic spectra of pyrazine without approximations inherent to analytical, including vibronic-coupling models. The Appendix provides derivations of perturbative expressions for linear and pump-probe spectra of arbitrary mixed states and for arbitrary laser pulse shapes.