We consider the averaging of local field-theoretic Poisson brackets in
the multi-dimensional case. As a result, we construct a local Poisson
bracket for the regular Whitham system in the multidimensional situation.
As can be shown, the special features of the space of modulated parameters
in higher dimensions lead to a different natural class of the averaged
brackets in comparison with the one-dimensional situation. We suggest
a direct procedure of construction of the bracket for the Whitham system
for $d > 1$ and discuss the conditions of applicability of the
corresponding scheme. At the end, we discuss canonical forms of the
averaged Poisson bracket in the multidimensional case.