I've now plowed through three usage manuals and it all seems to hinge on whether 'like' is being used as a conjunction or a preposition. If it's a conjunction, the following pronoun takes the nominative. If it's a preposition, the following pronoun takes the objective. And looking at the examples they give, I honestly can't tell the difference. To add to the confusion, the use of 'like' as a conjunction, while widespread, is frowned upon. And even if you accept it as a conjunction, several sources point out that using the objective pronoun after a conjunction is common usage and the 'correct' nominative often sounds ridiculously pedantic.

I think I'm going to give up now. Look! Cows!