In this paper, we study power bounded composition operators C_φ on weighted Dirichlet spaces D_α. As applications, we give the necessary and sufficient conditions for the composition operators to be Riesz operator on D_α, when C_φ is power bounded on D_β for some β between 0 and α. For α greater than 1, we completely characterize the Riesz composition operators on D_α. Some of the techniques developed in the paper are not new but lead to new results.