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. The criteria are evaluated numerically for tensile, compressive and shear tests. The extension of the LaRC05 stress based failure criteria to this class of 3D-woven composites is one possibility. This however, presents a number of challenges which are discussed. These challenges are related to the relative high stiffness in all directions, which produce excessively high shear components when projected onto potential off-axis failure planes. To circumvent these challenges, strain based criteria inspired by LaRC05 are formulated. Results show that strain based failure predictions for the simulated load cases are qualitatively more reasonable.