We report on the observation of nearly maximally entangled photon pairs from semiconductor quantum dots, without resorting to postselection techniques. We use GaAs quantum dots integrated on a patterned piezoelectric actuator capable of suppressing the exciton fine structure splitting. By using a resonant two-photon excitation, we coherently drive the biexciton state and demonstrate experimentally that our device generates polarization-entangled photons with a fidelity of 0.978(5) and a concurrence of 0.97(1) taking into account the nonidealities stemming from the experimental setup. By combining fine-structure-dependent fidelity measurements and a theoretical model, we identify an exciton spin-scattering process as a possible residual decoherence mechanism. We suggest that this imperfection may be overcome using a modest Purcell enhancement so as to achieve fidelities >0.99, thus making quantum dots evenly matched with the best probabilistic entangled photon sources.