Most string theory compactifications analyzed so far have as backgrounds a conformal field theory corresponding to a nonlinear sigma model with a Calabi-Yau target space, or some relatively classical background, possibly with orbifolding. But as long as the central charge of the Virasoro algebra is right, and worldsheet superconformal symmetry is respected, any other superconformal field theory will do just as well. Why haven't string theorists looked at such models more often?
Such models have been looked at in a fair amount of detail. For example, "Gepner" models are constructed by tensoring together superconformal minimal models. Asymmetric orbifolds which have the correct superconformal worldsheet symmetry have also been constructed. The problem is that if you break spacetime susy when you construct these theories directly as SCFTs you typically have tachyons and no good way to figure out the ground state of the system. If you preserve spacetime supersymmetry then many (all?) of the Gepner models turn out to describe special points in the moduli space of Calabi-Yau compactifications. So in this case at least, sigma-models with CY target end up being the more general construction. Some of the asymmetric orbifold constructions are not equivalent to CY as far as I know, but they are quite specialized and I don't know of any that have very realistic phenomenology. So basically the answer is that people tend to focus on the CY case because it is richer and you can do more with it. Of course there are probably some interesting constructions based on the SCFT approach you are asking about that have not been found yet. For example, I don't know of any Gepner model constructions of $G_2$ holonomy compactifications (I'd be happy to be corrected if these do in fact exist).