Such metrics describe what are known as Bianchi I universes after Luigi Bianchi who classified 3 dimensional Lie Algebras into eleven classes and showed that every such algebra is isomorphic to one and only one Lie Algebra on this list.
The Wikipedia page gives a brief summary of the classification, while - as @StephenG suggested - the Scholarpedia page is a little more thorough.
As can be clearly seen from the metric, Bianchi I spaces are in general anisotropic and thus not very good candidates to describe the universe (at least in cosmology), since the universe has been observed to be isotropic to great precision. Nevertheless, they have been studied - in particular, the Kasner metric, which describes for example anisotropic inflation.
Other mentions of such metrics can be found here and here.