In this research paper, a powerful numerical method is implemented to examine the free vibration analysis of functionally graded (FG) sandwich beams of variable cross-section. The core of the beam is considered to be made from homogenous isotropic material and the faces of the sandwich beam are considered to be made of FG materials. The effect of the shear strain is considered in the formulations based on the first-order shear deformation theory (FSDT). The governing canonical equations for the dynamic response of FG sandwich beams are derived with the aid of the canonically conjugate momentums for the first time. To solve these equations numerically the Complementary Functions Method (CFM) is implemented. The free vibration behavior of the symmetric and asymmetric FG sandwich beams is investigated for different boundary conditions. The presented efficient numerical method is validated with the available literature. Moreover, detailed parametric studies are conducted to demonstrate the influence of layers, geometric constants, material volume fraction indexes, and beam slenderness on the natural frequencies of FG sandwich beams.