We explore a variational approach to the finite-volume $N$-body problem. The general formalism for $N$ nonrelativistic spinless particles interacting with periodic pairwise potentials yields $N$-body secular equations. The solutions depend on the infinite-volume $N$-body wave functions. Given that the infinite-volume $N$-body dynamics may be solved by the standard Faddeev approach, the variational $N$-body formalism can provide a convenient numerical framework for finding discrete energy spectra in periodic lattice structures.