In this paper, we will compute asymptotically the eigenfrequencies for the in-plane vibrations of the general non-collinear Euler-Bernoulli beam equation with dissipative joints. Many different kinds of dampers are allowed, even within the same structure. This generalizes a previous result for collinear structures. Matrix techniques are used to combine asymptotic analysis with the wave propagation method. We will find that if the lengths of the beams are rational, there will be a finite number of “streams” of eigenfrequencies, and, like the collinear case, each lies asymptotically to a vertical line.