In the case of a globally hyperbolic spacetime, what you want is a smooth _Cauchy temporal function_ (the gradient is everywhere timelike, not just causal, and each level set is a Cauchy surface that is necessarily spacelike). That global hyperbolicity is also sufficient the the existence of a smooth temporal function was also shown by Bernal and Sanchez, in the followup [\[arXiv:gr-qc/0512095\]][1] to their original paper that you cite.

<s>I'm not sure if you are also interested in foliations of _non globally hyperbolic_ spacetimes. I'm not completely sure what the right conditions would be then.</s> See comment about _stable casuality_ below.


  [1]: http://arxiv.org/abs/gr-qc/0512095