The generalized Franck-Codon theory of the collinear dissociation of linear triatomic molecules is presented, including a proper description of the bending vibrations in the initial bound electronic state and of the rotational motions on both the initial and the final repulsive electronic surface. The nonseparable multidimensional bound-continuum Franck-Condon integrals are reduced to a rapidly convergent series of products of one dimensional integrals. Analytical expressions are derived for rotational and orbital angular momentum distributions of the products, for scalar coupling (as in predissociations), as well as parallel and perpendicular transitions (as in direct photodissociation). This fully quantum mechanical theory makes explicit the separate and interrelated roles played by angular momentum and energy conservation. The present work is applied in a separate paper to the photodissociation of ICN, and qualitative agreement with experiment is obtained.