Let $T^3=S^1\times S^1\times S^1$ be the standard three torus, and $\Gamma_m$ be the discrete group generated by $(e^{2\pi i/m},e^{2\pi i/m},e^{2\pi i/m})$, where each component $e^{2\pi i/m}$ acts on $S^1$ canonically.
Q I think that $T^3/\Gamma_m$ is a closed orientable flat manifold. But by the Wiki https://en.m.wikipedia.org/wiki/Flat_manifold, there are only 6 types of the orientable closed three flat manifold. Where goes wrong, could any one give a clue?
As you alluded to actions you write down are free and properly discontinuous actions, hence the quotients are naturally closed $3$-manifolds. The trick is that the quotients are all the $3$-torus. Note that quotienting the circle by itself by a cyclic subgroup gives the circle (one of first examples of covers we meet in topology), essentially the same reasoning shows that the quotient is just the $3$-torus.
With regards to fundamental group considerations, this corresponds to the fact that $\mathbb{Z}^{3}$ has non-surjective embeddings into itself (in this particular case just $(n_{1},n_{2},n_{3}) \mapsto m(n_{1},n_{2},n_{3})$).