In our previous studies we derived the exact closed structural theories for a set of plane regular and quasi-regular framed structure of girder type using the method of `gluing’. This allowed us to state and solve a wide range of problems of statics. In the paper presented, we extend this approach to problems of free vibrations of plane regular elastic girders of orthogonal structure. An element of the girder is a rectangle with two independent diagonal rods. In particular, this can be a girder with one diagonal rod in each element. These rods can be of one and the same or of different from element to element orientation. An exact general statement of the problem of free vibrations of the girder under consideration is realized in detail by exact reduction to a discrete eigenvalue problem. We examine some properties of the frequencies and modes of the girder oscillations. The theoretical results are illustrated by a numerical example.