We describe the construction, performance and first application of a spectrometer for measurements of frequency-dependent complex thermal diffusivities and effusivities. Knowledge of these quantities enables the complex dynamic specific heat and the thermal conductivity to be calculated. The technique uses the properties of thermal waves initiated at the surface of plate-like samples by the absorption of light with an oscillating intensity. A registration of the resulting temperature oscillations in the absorbance layer at the upper surface of the sample yields the effusivity, an analogous measurement at the bottom gives the diffusivity. The detection of both amplitudes and phases of the signals leads to complex quantities. Measurements encompass a frequency range from 10−2 to 102Hz. The apparatus was used in a study of the reversible surface melting of poly(ethylene oxide). Both the magnitude and the characteristic time range of this process could be determined.