Exact closed linear theories for plane elastic frameworks of regular and quasi-regular orthogonal structures have been developed earlier on the basis of the method of gluing and the method of initial parameters. Similar investigations were performed for three-dimensional regular frameworks of orthogonal structure composed of elementary cells of the shape of rectangular parallelepiped with one diagonal rod on each face. In this paper we generalize the previously obtained results to a wide class of regular and quasi-regular three-dimensional elastic frameworks of orthogonal structure. We pay main attention to the development of an exact closed linear theory for three-dimensional regular frameworks of orthogonal structure composed of elementary cells of the shape of rectangular parallelepiped with two diagonal rods on each face when these rods work independently of one another. Particular cases for such structures may be obtained by eliminating some rods from the basic elementary cell; as a result, we obtain a variety of regular and quasi-regular frameworks. The possibility for constructing particular cases for exact theories of this type is demonstrated in detail for one quasi-regular three-dimensional framework. The implementation of this particular theory is illustrated by an example of a compound rod with no internal joints; we consider rods of finite length, rods of infinite (in the both directions) length, and rods of semi-infinite length. For the cases mentioned, we obtain exact analytic solutions and perform calculation for some problems.