We introduce and characterize a new 3D cross-correlation algorithm, which relies on gradient-based iterative volume deformation. The algorithm, FOLKI3D, is the extension to 3D PIV of the approach introduced by Champagnat et al. 2011. It has a highly parallel structure and is implemented on GPU. Additionally to the gradient approach for displacement estimation, we implemented a high-order interpolation scheme (with cubic B-Splines) in the volume deformation step, at a reasonable computational cost. Performance tests on synthetic volumic distributions first allow to characterize the spatial transfer function of the algorithm, and to confirm the efficiency of this interpolator, comparable to that of standard image deformation methods in planar PIV. A second series of synthetic tests then investigates the response of FOLKI3D to sources of noise specific to the tomographic PIV context, i.e. ghost particles. Depending on the tests, the algorithm is found as efficient or more robust than the state-of-the-art. The gain brought by the high-order interpolation is also confirmed in a situation with a large number of ghosts, and different reconstructed particle shapes.