As a diclaimer, I'm still very new to the study of German. I'm assured that the proper translation of "I don't have a dog" is "ich habe keinen Hund" rather than "ich habe einen Hund nicht." The way that it was explained to me is that you're negating the noun rather than the verb.
I'm still slightly confused about the rule: in English, it's obviously perfectly acceptable to negate the verb. Why is this "wrong" in German in this case?