In [1,2], “gluing” and the method of initial parameters were used to construct new rigorous closed linear theories for plane regular elastic truss-type structures with a unit cell in the form of a rectangle with one or two noninteracting diagonal bars. In the present paper, the methodology set forth in [1,2] is applied to a quasi-regular plane elastic structure that differs from the structure studied in [1] in that the diagonal bars in adjacent cells run in different directions and generally have different rigidities. Alternative statements of the problems and some of their generalizations are given within the framework of the rigorous theory set forth for this structure. Application of the theory is illustrated in the cases of finite and infinite structures with one row of internal nodes, for which we give exact analytic solutions and results of specific calculations.