The parameter estimation problem for multivariate exponential sums is currently a very active field of research. One idea discussed in the last years is to solve multiple one dimensional parameter estimation problems and combine their results to get a solution of the higher dimensional problem. While these projection-based methods give reasonable results with only very few samples, they cannot efficiently use samples on a grid, which is the most common sampling set. In this paper we demonstrate how to extend the projection-based methods to cover parallel lines and hence samples on a grid.