The mechanical behavior of three-dimensional cross-linked random fiber networks composed from fibers of noncircular cross section characterized by two principal moments of inertia is studied in this work. Such fibers store energy in the axial deformation mode and two bending modes of unequal stiffness. We show that the torsional stiffness of fibers becomes important as it determines the relative contribution of the two bending modes to the overall deformation. The scaling of the small strain modulus with the network parameters is established. The large strain deformation of these structures is less sensitive to the shape of the cross section.