We present an exact method to model the free vibration of functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) beams with arbitrary boundary conditions based on first-order shear deformation elasticity theory. Five types of carbon nanotube (CNT) distributions are considered. The distributions are either uniform or functionally graded and are assumed to be continuous through the thickness of the beams. The displacements and rotational components of the beams are expressed as a linear combination of the standard Fourier series and several supplementary functions. The formulation is derived using the modified Fourier series and solved using the strong-form solution and the weak-form solution (i.e., the Rayleigh-Ritz method). Both solutions are applicable to various combinations of boundary constraints, including classical boundary conditions and elastic-supported boundary conditions. The accuracy, efficiency and validity of the two solutions presented are demonstrated via comparison with published results. A parametric study is conducted on the influence of several key parameters, namely, the L/h ratio, CNT volume fraction, CNT distribution, boundary spring stiffness and shear correction factor, on the free vibration of FG-CNTRC beams.