I read your question as the one about compact Cauchy surfaces in locally flat space times. Then the answer is negative: Take quotient of upper hyperboloid $H$ in $R^{2,1}$ (i.e. the hyperbolic plane) by a torsion free discrete cocompact subgroup $\Gamma$ in $SO(2,1)$. The Euler characteristic will be negative. Now, take the future light cone $C$ in $R^{2,1}$ and take the quotient $C/\Gamma$. This is your locally flat space-time, containing $H/\Gamma$ as a Cauchy hypersurface. This manifold is, of course, incomplete, but, if I remember correctly, Geoff Mess proved that you cannot have a compact Cauchy hypersurface in a complete locally flat Lorentzian manifold.
Edit: On the other hand, there are topological obstructions for existence of locally flat Lorenztian metrics. For instance, suppose that $M$ is a compact n-dimensional simply-connected manifold which does not immerse in $R^{n+1}$. For instance, $CP^2$ is an example of such manifold. Then $M\times R$ does not admit a locally flat Lorentzian metric, even incomplete one. (This follows because the developing map to $R^{n,1}$ of such a structure would yield an immersion to $R^{n,1}$.
Talking about tiles: My guess is that you are actually trying to construct complete locally flat Lorentzian metrics. It is known (proved by Geoff Mess) that such metrics do not exist on manifolds of the form $M\times R$, where $M$ is compact and hyperbolic. I will check, by now there is probably a reasonably good description of $M$'s for which such metric exists. On the other hand, if you take $M$ which is a noncompact surface, then such a metric does exist; these are so called Margulis space-times. One even has some nice description of their fundamental domains (tiles) due to Todd Drumm. The trick is that the tiles are not convex. (Google "crooked planes" to learn more about them.) I am not sure if $M$ will be a Cauchy hypersurface in this case, I would have to check.
See e.g. here for a somewhat dated survey.