A powerful semi-analytical method is developed to investigate the impact of tough inclusions on crack paths and the resulting effective fracture properties. Using a perturbative approach of fracture mechanics, this method allows to account for out-of-plane excursions of cracks resulting from the discontinuity of toughness at the inclusion/matrix interface, by applying a generalized maximum-energy-release-rate criterion. For a sufficiently large toughness mismatch between the matrix and the inclusions, the crack bypasses the obstacles, thus reducing the inclusion-induced reinforcement of the material. For spherical inclusions, obstacles of fracture energy larger than 3.85 times that of the matrix become ineffective to further reinforce the material, because of their systematic bypass. The role played by the shape of the inclusions is also investigated; inclusions of non-spherical shape may reduce the reinforcement through three-dimensional collective effects emerging from the coupling between the in-plane and out-of-plane components of the crack front perturbation. Finally we show that inclusion bypass not only limits crack bowing but can also prevent crack bridging, for inclusions short enough in the direction orthogonal to the mean fracture plane. Thus this study provides a quantitative picture of the interaction mechanisms between a crack front and tough inclusions, which paves the way to the future microstructural design of brittle solids with improved resistance to failure.