Motivated by the recent measurements from the VEPP-3 electron storage ring, we investigate the tensor target polarization asymmetries T2M (M = 0, 1, 2) in the reaction γ d → π0d with a particular interest in the effect of the intermediate ηNN three-body approach. This approach is based on realistic separable representations of the driving two-body interaction in the πN, ηN, and NN subsystems. It is shown that the influence of rescattering effects in the intermediate state on the tensor target spin asymmetries is sizable at extreme backward pion angles. At forward angles, the contribution from the pure impulse approximation is dominated and the spin asymmetries show very little influence of rescattering effects. The sensitivity of results to the elementary pion photoproduction operator and to the NN potential model adopted for the deuteron wave function is investigated, and considerable dependences are found. The predicted spin asymmetries are also compared with available experimental data, and a satisfactory agreement with the recent data from VEPP-3 is obtained at photon energies below 400 MeV. At higher energies, the calculated spin asymmetries slightly underestimate the data.