  Fully relativistic approach is applied to 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 $\alpha Z$. The
calculations are performed in the range $1 \leqslant Z \leqslant 20$ for the
$g$ factors of $1s$, $2s$, $2p_{1/2}$, and $2p_{3/2}$ states. The $\alpha
Z$-dependence of the nontrivial QED recoil contribution as a function of $Z$ is
studied.
