In this study, torsional wave propagation is investigated in a rod that are made of one and two dimensional functionally graded material. Firstly, the governing equations of the wave propagation in the functionally graded cylinder derived in polar coordinate. Secondly, finite difference method is used to discretize the equations. The Von Neumann stability approach is used to obtain the time step size. Two states are assumed for material distribution, in first state it’s considered that the material variation occurred only in radial direction(Ti6A14V and Al2O3) and in second state the material properties vary in radial and length directions(BN, Al 1100, Ti6A14V and Al2O3). Moreover, the effect of cutoff frequency and boundary condition in wave propagation is studied. The results was validated by comparing the analytical and numerical solutions for an isotropic rode subjected to a torsional impulsive load. The results show that the torsional wave propagation in FGM rod evidently effects by material composition variation.