  We present a method for reconstructing the phonon relaxation time
distribution $\tau_{\omega}= \tau (\omega)$ (including polarization) in a
material from thermal spectroscopy data. The distinguishing feature of this
approach is that it does not make use of the effective thermal conductivity
concept and associated approximations. The reconstruction is posed as an
optimization problem in which the relaxation times $\tau_{\omega}= \tau
(\omega)$ are determined by minimizing the discrepancy between the experimental
relaxation traces and solutions of the Boltzmann transport equation (BTE) for
the same problem. The latter may be analytical, in which case the procedure is
very efficient, or numerical. The proposed method is illustrated using Monte
Carlo solutions of thermal grating relaxation as synthetic experimental data.
The reconstruction is shown to agree very well with the relaxation times used
to generate the synthetic Monte Carlo data and remains robust in the presence
of uncertainty (noise).
