Spatial averaging is a new approach for sampling rare-event problems. The approach modifies the importance function which improves the sampling efficiency while keeping a defined relation to the original statistical distribution. In this work, spatial averaging is applied to multidimensional systems for typical problems arising in physical chemistry. They include (I) a CO molecule diffusing on an amorphous ice surface, (II) a hydrogen molecule probing favorable positions in amorphous ice, and (III) CO migration in myoglobin. The systems encompass a wide range of energy barriers and for all of them spatial averaging is found to outperform conventional Metropolis Monte Carlo. It is also found that optimal simulation parameters are surprisingly similar for the different systems studied, in particular, the radius of the point cloud over which the potential energy function is averaged. For H(2) diffusing in amorphous ice it is found that facile migration is possible which is in agreement with previous suggestions from experiment. The free energy barriers involved are typically lower than 1 kcal/mol. Spatial averaging simulations for CO in myoglobin are able to locate all currently characterized metastable states. Overall, it is found that spatial averaging considerably improves the sampling of configurational space.