Modern agriculture produces a very large amount of agricultural waste that remains unused. The use as a reinforcer of these renewable resources for the realization of composite materials, and the finding of useful industrial applications, constitutes or provokes the groups of researchers in this field. The study conducted in this article falls in this direction. Composites were fabricated with the chopped wheat straw reinforcement and epoxy resin matrix or hybrid resins with 50% and 70% Dammar volume proportions. Some mechanical properties of this type of composite materials were studied based on tensile strength, SEM analysis, water absorption/loss, vibration behavior and compression strength. The strength–strain and strain–strain diagrams, the modulus of elasticity, the breaking strength and the elongation at break were obtained. Compared to the epoxy resin composition, those with 50 and 70% Dammar, respectively, have a 47 and 55% lower breaking strength and a 30 and 84% higher damping factor, respectively. Because the values of these mechanical properties were limited, and in practice superior properties are needed, sandwich composites were manufactured, with the core of previously studied compositions, to which the outer faces of linen fabric were applied. These composites were applied to the bend (in three points), obtaining the force–deformation diagrams. The obtained properties show that they can be used in construction (paneling, shells, etc.), or in the furniture industry.