Examples of nontrivial (that is, non-Cartesian product) multidimensional discrete variable representation (DVR) basis sets are presented that are generalizations of sinc functions in one dimension. Their use in solving quantum problems in two dimensions is illustrated. Unlike all standard (one-dimensional) examples of DVR bases, these bases cannot be created by dividing out the roots of a generating function. It is argued that the difficulty of constructing nontrivial, multidimensional DVR bases is due to the restrictive nature of the DVR conditions, which cannot be satisfied on most subspaces of wave functions. The bases considered in this paper, however, are invariant under translations on a lattice in n-dimensional space, and the properties of the Abelian group of lattice translations allow the DVR conditions to be satisfied. More generally, the question of the relation between group theory and the conditions necessary to qualify a set of basis functions as a DVR basis is considered. It is shown how to construct orthonormal states that are related by the action of some group, and, in the case of Abelian groups, the extra conditions required to qualify the basis as a DVR set are discussed.