Diagonal CFTs such as Liouville theory can be solved analytically using the conformal bootstrap method. Here we extend this approach to non-diagonal CFTs, i.e. CFTs whose primary fields have nonzero conformal spins. \par Assuming generic values of the central charge, we find constraints on the non-diagonal sector of the spectrum, and then derive and solve the equations that determine the three- and four-point structure constants of non-diagonal theories. We will discuss how these structure constants can be expressed in terms of those of the diagonal case, and then test our results by performing numerical checks of crossing symmetry of four-point functions in a non-rational limit of D-series minimal models.