Lowenheim-Skolem is going to give you trouble, unless "coherently-thinkable" is meant of as a subtantive restriction. You might be able to enumerate finitely-axiomatisable models, up to isomorphism, up to aleph-w, if you limit yourself to k-categorical theories, for k < aleph-w, though. Then you could use Will's strategy and enumerate axioms.

Edit: I realised I'm being pointlessly obscure.

The Upwards Lowenheim-Skolem means that, for every set of axioms in your list, you'll have multiple (non-isomorphic) models.

You might avoid this if "coherantly thinkable" was taken to mean "of small cardinality".

If you didn't enjoy this restriction, you could, for any given set of axioms, enumerate the k-categorical models of that set of axioms - or at least enumerate the models of whose cardinality can be expressed as 2^2^...2^w, for some finite number of 2's. This is because k-categoriciticy means you'll only have one model of each cardinality, up to isomorphism.

So then you just enumerate all the possible countable combinations of axioms, and you have an enumeration of all countably axiomatisable, k-categorical, models.