This paper presents a new method for calculating linear cosmic microwave background (CMB) anisotropy spectra based on integration over sources along the photon past light cone. The method separates the temperature anisotropy into a geometrical term and a source term. The geometrical term, which oscillates rapidly, does not depend on the cosmological model and can be precomputed. The source term, which depends on the model and is generally the most time-consuming part of the calculation, is evaluated at a small number of points. This approach significantly reduces computational time compared to standard methods, achieving a two-order-of-magnitude reduction in CPU time. The method is particularly useful for accurate cosmological parameter determination from future CMB observations. It allows for fast and accurate calculations of CMB anisotropies and polarization spectra without approximations, making it suitable for parameter space searches. The method is implemented by solving the equations in integral form, which separates the source and geometrical contributions. This approach enables efficient computation of the CMB power spectrum by reducing the number of differential equations to be solved. The method is validated against existing calculations and shows high accuracy, with relative errors below 1% for most models. The computational efficiency and accuracy make this method a valuable tool for future CMB experiments.This paper presents a new method for calculating linear cosmic microwave background (CMB) anisotropy spectra based on integration over sources along the photon past light cone. The method separates the temperature anisotropy into a geometrical term and a source term. The geometrical term, which oscillates rapidly, does not depend on the cosmological model and can be precomputed. The source term, which depends on the model and is generally the most time-consuming part of the calculation, is evaluated at a small number of points. This approach significantly reduces computational time compared to standard methods, achieving a two-order-of-magnitude reduction in CPU time. The method is particularly useful for accurate cosmological parameter determination from future CMB observations. It allows for fast and accurate calculations of CMB anisotropies and polarization spectra without approximations, making it suitable for parameter space searches. The method is implemented by solving the equations in integral form, which separates the source and geometrical contributions. This approach enables efficient computation of the CMB power spectrum by reducing the number of differential equations to be solved. The method is validated against existing calculations and shows high accuracy, with relative errors below 1% for most models. The computational efficiency and accuracy make this method a valuable tool for future CMB experiments.