We propose a new method to recover the cosmological initial conditions of the presently observed galaxy distribution, which can serve to run constrained simulations of the Local Universe. Our method, the Reverse Zeldovich Approximation (RZA), can be applied to radial galaxy peculiar velocity data and extends the previously used Constrained Realizations (CR) method by adding a Lagrangian reconstruction step. The RZA method consists of applying the Zeldovich approximation in reverse to galaxy peculiar velocities to estimate the cosmic displacement field and the initial linear matter distribution from which the present-day Local Universe evolved.We test our method with a mock survey taken from a cosmological simulation. We show that the halo peculiar velocities at z = 0 are close to the linear prediction of the Zeldovich approximation, if a grouping is applied to the data to remove virial motions. We find that the addition of RZA to the CR method significantly improves the reconstruction of the initial conditions. The RZA is able to recover the correct initial positions of the velocity tracers with a median error of only 1.36 Mpc/h in our test simulation. For realistic sparse and noisy data, this median increases to 5 Mpc/h. This is a significant improvement over the previous approach of neglecting the displacement field, which introduces errors on a scale of 10 Mpc/h or even higher. Applying the RZA method to the upcoming high-quality observational peculiar velocity catalogues will generate much more precise constrained simulations of the Local Universe.