The solution of linear inverse heat conduction problems (IHCP) for the reconstruction of time-dependent, spatially unknown heat fluxes on the boundaries of two- and three-dimensional geometries from several temperature measurements is presented. The solution method is based on the interpretation of the IHCP in the frequency domain. We emphasize the extension of the method to large-scale problems. In particular, we consider the parameterization of the unknown heat fluxes with respect to the number of sensors and their positions and the reduction and stable inversion of the large linear state-space models obtained from semidiscretization of the heat conduction equation. The influence of each of the steps and the choice of their tuning parameters on the quality of the estimated heat flux are discussed. Two- and three-dimensional examples show the good performance of the method with regard to time and space resolution in the presence of noisy measurements.