We have investigated the dynamics of bright solitons in a spin-orbit coupled spin-1 Bose-Einstein condensate analyt-ically and numerically.By using the hyperbolic sine function as the trial function to describe a plane wave bright soliton with a single finite momentum,we have derived the motion equations of soliton's spin and center of mass,and obtained its exact analytical solutions.Our results show that the spin-orbit coupling couples the soliton's spin with its center-of-mass motion,the spin oscillations induced by the exchange of atoms between components result in the periodical oscillation of center-of-mass,and the motion of center of mass of soliton can be viewed as a superposition of periodical and linear motions.Our analytical results have also been confirmed by the direct numerical simulations of Gross-Pitaevskii equations.