This paper is concerned with estimating unknown multidimensional frequencies from linear compressive measurements. This is accomplished by employing the recently proposed atomic norm minimization framework to recover these frequencies under a sparsity prior without imposing any grid restriction on these frequencies. To this end, we give a rigorous derivation of an iterative scheme called alternating direction of multipliers method, which is able to incorporate multiple compressive snapshots from a multi-dimensional superposition of complex harmonics. The key result here is how to formulate the objective function minimized by this scheme and its partial derivatives, which become hard to manage if the dimensionality of the frequencies is larger than 1. Moreover we demonstrate the performance of this approach in case of 3D line spectral estimation and 2D DOA estimation with a synthetic antenna array.