We consider the pure-power defocusing nonlinear Klein-Gordon equation, in the H-1-subcritical case, posed on the product space R-d X T, where T is the one-dimensional flat torus. In this framework, we prove that scattering holds for any initial data belonging to the energy space H(1)x L-2 for 1 <= d <= 4. The strategy consists in proving a suitable profile decomposition theorem on the whole manifold to pursue a concentration-compactness and rigidity method along with the proofs of (global in time) Strichartz estimates.