We consider a one-dimensional nonlinear periodic structure that supports the effect of nonreciprocity of elastic wave propagation. It is known that microstructured media have the ability to control wave propagation due to the internal microstructure. Elastic linear lattices consisting of point masses and elastic non-inertial ligaments exhibit dispersive and filtering properties, even if they are homogeneous. To exploit this advantage, it is important to understand the dynamic properties of the structure. The dispersion relation results for different models considered in this paper (monoatomic, diatomic systems with the same and different stiffness) show that the new nonlinear elastic periodic structure has obvious advantages over other types of periodic structures, including greater sensitivity to flexibility in band gap manipulation. The dynamic behavior of the elastic system is investigated analytically and compared with numerical solutions. The results for the dispersion relation model demonstrated obvious advantages related to the manipulation of elastic wave propagation. Based on the obtained results, the design of non-reciprocal device is introduced by including linear and non-linear circuits. The specific non-reciprocal effect for transmitted waves due to coupling effect is verified based on the theoretical concept supported by the calculation of two mutual works between applied forces.