We present a new method for calculating linear cosmic microwave background (CMB) anisotropy spectra based on integration over sources along the photon past light cone. In this approach the temperature anisotropy is -written as a time integral over the product of a geometrical term and a source term. The geometrical term is given by radial eigenfunctions, which do not depend on the particular cosmological model. The source term can be expressed in terms of photon, baryon, and metric perturbations, all of which can be calculated using a small number of differential equations. This split clearly separates the dynamical from the geometrical effects on the CMB anisotropies. More importantly, it allows us to significantly reduce the computational time compared to standard methods. This is achieved because the source term, which depends on the model and is generally the most time-consuming part of calculation, is a slowly varying function of wavelength and needs to be evaluated only in a small number of points. The geometrical term, which oscillates much more rapidly than the source term, does not depend on the particular model and can be precomputed in advance. Standard methods that do not separate the two terms require a much higher number of evaluations. The new method leads to about 2 orders of magnitude reduction in CPU time when compared to standard methods and typically requires a few minutes on a workstation for a single model. The method should be especially useful for accurate determinations of cosmological parameters from CMB anisotropy and polarization measurements that will become possible with the next generation of experiments. A program implementing this method can be obtained from the authors.
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