Three dimensional (3D) fibre-reinforced composites have shown weight efficient strength and stiffness characteristics as well as promising energy absorption capabilities. In the considered class of 3D-reinforcement, vertical and horizontal weft yarns interlace warp yarns. The through-thickness reinforcements suppress delamination and allow for stable and progressive damage growth in a quasi-ductile manner. With the ultimate goal of developing a homogenised computational model to predict how the material will deform and eventually fail under loading, this work proposes candidates for failure initiation criteria. It is shown that the extension of the LaRC05 stress-based failure criteria for unidirectional laminated composites, to this class of 3D-reinforced composite presents a number of challenges and leads to erroneous predictions. Analysing a mesoscale representative volume element does however indicate, that loading the 3D fibre-reinforced composite produces relatively uniform strain fields. The average strain fields of each material constituent are well predicted by an equivalent homogeneous material response. Strain based criteria inspired by LaRC05 are therefore proposed. The criteria are evaluated numerically for tensile, compressive and shear tests. Results show that their predictions for the simulated load cases are qualitatively more reasonable.