Ethylene–(vinyl alcohol) (EVOH) copolymer/organo-modified montmorillonite (oMMT) composites were investigated. Composites with two different percentages by weight of oMMT were prepared using a melt-extrusion procedure in a twin-screw extruder. Films made of EVOH and EVOH/oMMT composites were prepared in a cast-film extrusion line. The processing conditions allow inducing strong orientation of polymer chains. The study is to evaluate both the effect of the clay particles on the mechanical properties and the influence of the particles to hinder the process-induced orientation and how it affects the mechanical results. The fracture behavior was analyzed by using the essential-work-of-fracture (EWF) method. Fracture characterization was carried out for the two main processing directions: melt flow direction (MD) and transverse direction (TD). Fractographic observations were made using scanning electron microscopy (SEM). The tensile test results indicated good compatibility between EVOH and oMMT. In addition, the fracture tests showed the influence of the clay particle arrangement on the fracture behaviour, showing an increase in the specific essential work of fracture, we, which was attributed to the EVOH–oMMT interaction. The clay particles hindered the orientation of the polymer chains as a consequence of the limitations on the plastic flow, promoting different trends in the plastic term, βwp, depending on the test direction. This behaviour was explained by the size of the plastic zone and the restrictions to the EVOH plastic flow originated by the clay particles. It was revealed that polymer chains oriented in the MD direction show high plastic deformation, increasing the βwp term. The fracture of the films was not possible to evaluate in the TD direction. In this sense, the EWF method is shown to be a very useful and sensitive tool for the analysis of structure–property relationships in polymer–organo-clay composites.