  In this paper, we determine, in the case of the Laplacian on the flat
three-dimensional torus $(\mathbb{R}/\mathbb{Z})^3$, all the eigenvalues having
an eigenfunction which satisfies the Courant nodal domains theorem with
equality (Courant-sharp situation). Following the strategy of {\AA}. Pleijel
(1956), the proof is a combination of an explicit lower bound of the counting
function and a Faber-Krahn-type inequality for domains on the torus, deduced
as, in the work of P. B\'erard and D. Meyer (1982), from an isoperimetric
inequality. This inequality relies on the work of L. Hauswirth, J. Perez, P.
Romon, and A. Ros (2004) on the periodic isoperimetric problem.
