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Any closed simply-connected 4-manifold $M$ and its exotic copy $M'$ are h-cobordant by a theorem of Wall. Thus $M\times S^1$ is h-cobordant to $M'\times S^1$ (as one can extend the previus h-cobordsim trivially on the $S^1$ component) which is in fact trivial by high dimensional s-cobordism theorem which says that such a cobordism is trivial if Whitehead torsion of $\pi_1(M\times S^1)$ vanish (https://en.wikipedia.org/wiki/H-cobordism#The_s-cobordism_theorem). So they are in fact diffeomorphic as $Wh(\pi_1(M\times S^1))= Wh(\mathbb Z)=0$ by a result of Bass (http://www.numdam.org/item/?id=PMIHES_1964__22__61_0). And similarly one can generate more examples I guess in other dimensions.