Additive manufacturing enables the fabrication of objects embedding metamaterials. By creating fine-scale structures, the object's physical properties can be graded (e.g. elasticity, porosity), even though a single base material is used for fabrication. Designing the fine and detailed geometry of a metamaterial while attempting to achieve specific properties is difficult. In addition, the structures are intended to fill comparatively large volumes, which quickly leads to large data structures and intractable simulation costs. Thus, most metamaterials are defined as periodic structures repeated in regular lattices. The periodicity simplifies modeling, simulation, and reduces memory costs -- however it limits the possibility to smoothly grade properties along free directions. In this work, we propose a novel metamaterial with controllable, freely orientable, orthotropic elastic behavior -- orthotropy means that elasticity is controlled independently along three orthogonal axes, which leads to materials that better adapt to uneven, directional load scenarios, and offer a more versatile material design primitive. The fine-scale structures are generated procedurally by a stochastic process, and resemble a foam. The absence of global organization and periodicity allows the free gradation of density, orientation, and stretch, leading to the controllable orthotropic behavior. The procedural nature of the synthesis process allows it to scale to arbitrarily large volumes at low memory costs. We detail the foam structure synthesis, analyze and discuss its properties through numerical and experimental verifications, and finally demonstrate the use of orthotropic materials for the design of 3D printed objects.