The recently described equivalence between the alignment of two proteins and a conformation of a lattice chain on a two-dimensional square lattice is extended to multiple alignments. The search for the optimal multiple alignment between several proteins, which is equivalent to finding the energy minimum in the conformational space of a multi-dimensional lattice chain, is studied by the Monte Carlo approach. This method, while not deterministic, and for two-dimensional problems slower than dynamic programming, can accept arbitrary scoring functions, including non-local ones, and its speed decreases slowly with increasing number of dimensions. For the local scoring functions, the MC algorithm can also reproduce known exact solutions for the direct multiple alignments. As illustrated by examples, both for structure- and sequence-based alignments, direct multi-dimensional alignments are able to capture weak similarities between divergent families much better than ones built from pairwise alignments by a hierarchical approach.