A fully relativistic approach is applied to the evaluation of the nuclear recoil effect on the bound-electron g factor in hydrogenlike ions to first order in the electron-to-nucleus mass ratio m/M and to all orders in αZ. The calculations are performed in the range 1≤Z≤20 for g factors of the 1s, 2s, 2p1/2, and 2p3/2 states. The αZ dependence of the nontrivial QED recoil contribution as a function of Z is studied.