In [1,2], a “gluing” procedure and the initial parameter method were used to construct new rigorous closed linear theories for plane regular truss-type elastic structures with unit cells in the form of rectangles with one and two noninteracting diagonal bars. In this 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 successive cells along one direction have different directions and, generally speaking, different rigidities. Alternative problem statements with certain generalizations are given within the framework of the rigorous theory set forth for this structure. Application of the theory is illustrated in examples of finite and unbounded structures with one row of internal nodes, for which exact analytic solutions are obtained and results of specific calculations presented.