A method of constructing explicit formulae for Green's matrix function of the time-dependent Maxwell system for a homogeneous non-dispersive dielectric with a general form of the anisotropy is described. This method consists of two parts. In the first part a matrix function depending on the time variable and three Fourier parameters is determined. This matrix function, being the Fourier image of Green's matrix function with respect to three space variables, is constructed in an explicit form. The second part of the method is the inverse Fourier transform of the determined matrix function. The robustness and advantages of this method are confirmed by numerical experiments. Examples of simulations of electromagnetic fields in anisotropic crystals are presented.