The free vibration and modal power flow of a functionally graded material (FGM) beam with an open crack are studied. The crack is simulated by using the massless-rotational spring model. The natural frequencies and corresponding modal shapes of the cracked beam are obtained by the wave propagation method. A modal power flow formula of the FGM beam is deduced by Bernoulli-beam theory. A detailed parametric study is conducted to show the influences of crack location, crack depth, material property gradient, and boundary condition on the modal power flow characteristics based on a modal power flow damage index. Numerical examples show that the damage index based on the modal power flow can effectively identify the crack in the FGM beam, which provides the basis for the future study on the modal power flow based damage detection of functionally graded material structures.