The approximation of vibrational adiabaticity in curvilinear natural collision coordinates is investigated for tunneling in three-atom collinear reactions. A validity criterion is derived which limits the adiabatic approximation to systems with small reaction-path curvature. A general formalism is developed for systems which satisfy this criterion. A one-dimensional Schrödinger equation is proposed which is sufficiently flexible so as to be adaptable to many different models of tunneling. We present three new methods for including reaction-path curvature effects on multidimensional tunneling in reactive systems: a method based on a quantum mechanical vibrational average (VA) over degrees of freedom transverse to the minimum-energy path; a method (called DA for dynamical-path vibrational-average) that includes internal centrifugal effects in the description of the transverse vibrational motion (in this method the vibrational average is approximated as a quantal vibrational average about the dynamical path along which the Born-Oppenheimer force cancels the internal centrifugal force); and a semiclassical optical potential (SOP) method based on the Feshbach formalism translated into an adiabatic representation with reaction-path curvature providing the coupling mechanism between the explicit and implicit spaces. These models are compared, both formally and numerically, to each other and to four other methods that have been proposed previously, including the small-curvature (SC) approximation that we have proposed in a recent communication. The VA and SOP methods are shown to provide generalizations of phase average (PA) and second-order (SO) methods proposed earlier by Miller and co-workers. The difference is that vibrations are treated quantum mechanically in the VA and SOP methods but classically and harmonically in the PA and SO methods; the quantum mechanical methods have the advantage that anharmonicity can be included more straightforwardly. The DA, SO, and SOP methods, although they include internal centrifugal effects more fully than the VA and PA methods, do not offer significant improvement in accuracy. The numerical results clearly support the physical interpretation of the collapse of the vibrational wave function about a least-action path. The most successful methods are the Marcus-Coltrin path (MCP) and SC approximations. These methods, especially the SC approximation because it is more general, are recommended for future applications.