In this paper, a new method for the estimation of the parameters of multidimensional ($R$-D) harmonic and damped complex signals in noise is presented. The problem is formulated as $R$ simultaneous sparse approximations of multiple 1-D signals. To get a method able to handle large size signals while maintaining a sufficient resolution, a multigrid dictionary refinement technique is associated to the simultaneous sparse approximation. The refinement procedure is proved to converge in the single $R$-D mode case. Then, for the general multiple modes case, the signal tensor model is decomposed in order to handle each mode separately in an iterative scheme. The proposed method does not require an association step since the estimated modes are automatically ``paired''. We also derive the Cram\'er-Rao lower bounds of the parameters of modal $R$-D signals. The expressions are given in compact form in the single tone case. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed method.