Suppose n observations are to be taken within a compact region, where the objective is to estimate the mean level of a multidimensional stationary process using the ordinary sample mean as the estimator. Simulated annealing is used to search for optimal (variance minimizing) designs for the case when the observations are correlated. The results give insight into the sensitivity of optimal designs to the strength and nature of the correlation present, extending and reinforcing previous results for the one-dimensional case. An important outcome is that designs which space out the sampling locations evenly are optimal if the correlation is low.