The main purpose of this work is to present an efficient approximate approach for solving linear systems of fractional integro-differential equations based on a new application of Taylor expansion. Using the extit{m}th-order Taylor polynomial for unknown functions and employing integration method the given system of fractional integro-differential equations will be converted into a system of linear equations with respect to unknown functions and their derivatives. The solutions of this system yield the approximate solutions of fractional integro-differential equations system. The Riemann-Liouville fractional derivative is applied in calculations. An error analysis is discussed as well. The accuracy and the efficiency of the suggested method is illustrated by considering five numerical examples.