We report on the theoretical derivation and experimental observation of spatiotemporal modulation instability (MI) of a coherent light beam in noninstantaneous nonlinear media. We obtain analytically the MI growth rate as a function of the spatial and temporal frequencies of the perturbation and the material response time. In the experiment, we observe that the varying speed of the MI patterns increases with the decreased material response time. We also observe that increasing the material response time can arrest the MI, agreeing with our theoretical derivation.