The cumulative reaction probability (CRP) is a gross characteristic of rearrangement collision processes defining the reaction rate constant. This paper presents a complete development of the approach to the theory of CRP that we have recently proposed [Phys. Rev. Lett. 80, 41 (1998)]. In the core of this approach lies an alternative expression for CRP in terms of the outgoing wave Green's function which is formally equivalent to the Miller's definition of this quantity in terms of the scattering matrix [J. Chem. Phys. 62, 1899 (1975)] and to the Miller-Schwartz-Tromp formula [J. Chem. Phys. 79, 4889 (1983)], but is direct, in contrast to the former, and more suitable for practical calculations than the latter. Furthermore, our approach rests on solid grounds of time-independent quantum scattering theory and provides an appealing competitive alternative to the absorbing potential formulation given by Seideman and Miller [J. Chem. Phys. 96, 4412 (1992); 97, 2499 (1992)]. Ideologically, it is close to the approach considered earlier for a one-dimensional model by Manolopoulos and Light [Chem. Phys. Lett. 216, 18 (1993)], but is formulated from scratch for realistic systems with many degrees of freedom. The strongest point of our approach is that its final working formulas are expressed in terms of the Wigner-Eisenbud $\mathcal{R}$ matrix, so they can be easily implemented on the basis of many existing quantum scattering codes. All these features are discussed and illustrated by calculations of the CRP and reaction eigenprobabilities for two prototypical light atom transfer reactions in heavy-light-heavy triatomic systems in three dimensions for zero total angular momentum.