It has been shown recently that in order for real-time correlation functions obtained from centroid molecular dynamics (CMD) simulations to be directly related, without further approximations, to the corresponding quantum correlation functions, one of the operators should be linear in the position and/or momentum [Jang and Voth, J. Chem. Phys. 111, 2357 (1999)]. Standard reaction rate theory relates the rate constant to the flux-Heaviside or the flux-flux correlation functions, which involve two nonlinear operators and therefore cannot be calculated via CMD without further approximations. We present an alternative, and completely equivalent, reaction rate theory which is based on the position-flux correlation function. The new formalism opens the door to more rigorously using CMD for the calculation of quantum reaction rate constants in general many-body systems. The new method is tested on a system consisting of a double-well potential bilinearly coupled to a harmonic bath. The results obtained via CMD are found to be in good agreement with the numerically exact results for a wide range of frictions and temperatures.