In this paper, a three-dimensional time-dependent nonlinear Riesz space-fractional reaction-diffusion equation is considered. First, a linearized finite volume method, named BDF-FV, is developed and analyzed via the discrete energy method, in which the space-fractional derivative is discretized by the finite volume element method and the time derivative is treated by the backward differentiation formulae (BDF). The method is rigorously proved to be convergent with second-order accuracy both in time and space with respect to the discrete and continuous $L^2$ norms. Next, by adding high-order perturbation terms in time to the BDF-FV scheme, an alternating direction implicit linear finite volume scheme, denoted as BDF-FV-ADI, is proposed. Convergence with second-order accuracy is also strictly proved under a rough temporal-spatial stepsize constraint. Besides, efficient implementation of the ADI method is briefly discussed, based on a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic systems. Numerical experiments are presented to support the theoretical analysis and demonstrate the effectiveness and efficiency of the method for large-scale modeling and simulations.