This paper proposes an approach for the exact solution of the transverse free vibration of exponential axially functionally graded material (AFGM) Timoshenko beams with concentrated tip masses and general boundary conditions. Initially, by utilizing the governing equilibrium equations of a Timoshenko beam, the main differential equation for the free vibration of the AFGM Timoshenko beam is obtained. Then, the beam deformation function is achieved by solving the governing equation of the beam vibration exactly. Subsequently, by applying the boundary conditions, the constant coefficient matrix of the beam becomes available. By making the determinant of the constant coefficients zero, the characteristic equation of the system and consequently, the beam's natural frequencies are obtained. It is noteworthy that the final relation is presented so that it can be used to find the exact frequencies of homogeneous and inhomogeneous Euler-Bernoulli, Rayleigh, and shear beams, too. Numerical examples demonstrate the accuracy of the results obtained by the proposed method. In the next section of the paper, the effect of the exponential gradient index, elastic end supports, concentrated tip mass, rotational inertia of the concentrated mass, and thickness-to-length ratio on the natural frequencies and mode shapes of the Timoshenko beam is investigated. The results show that the exponential gradient index, boundary conditions, concentrated tip mass, and thickness-to-length ratio play an influential role in the dynamic behavior of AFGM beams.