Quasifree neutral-pion photoproduction from the deuteron is studied in the energy region near threshold by considering np- and π^0N-rescattering in the final state (FSI). Total and differential cross sections for the inclusive reaction d(γ, π^0)np have been predicted. Sizeable effects from both the np- and π^0N-FSI are found close to threshold. With increasing photon energy, we found that np-rescattering, in contrast to π^0N-rescattering, is important and leads to a big reduction of the cross sections at forward pion angles. The extracted cross sections are compared with the available experimental data and other theoretical models, and discrepancies are evident, perhaps this is a signal of two-body effects in the photoproduction amplitude or a different elementary production operator.