Here's a relatively simple "restriction" I've been toying with: Cloning doesn't work on humans.

This does a couple things for you:
1. It restricts "primary" characters from being clones so the audience doesn't feel like you're jerking them around with fake deaths, or clone deaths or having to be suspect whether the main characters are themselves or not.
2. If it's only the humans that can't be cloned, it gives the Empire the rationale they need to be distrustful of non Human races. All important leadership roles would need to be held by Humans since they are beyond suspicion of being replaced by clones.
3. If it's a single race that can be cloned: That means anytime you see that race, it's probably a clone. If the Clone Wars are destructive enough, then some galactic leaders would approve of complete genocide of that race of creatures if it means "restoring peace" to the rest of the galaxy.

I'll add one more and this is where my ideas get most unpopular: (some of it is here: http://originaltrilogy.com/forum/topic.cfm/The-Emperors-New-Clones/topic/11008/ )

Perhaps the human clones "don't work" because they "have no mind." Their bodies are completely functional, but their brains just don't work. They simply lie as if in a coma. But Palpatine, seeking immortality through manipulations of the Dark Side, is also using the clones- not to build an army, but to create an infinite supply of Palpatine bodies which he can use to transfer his dark soul into. So this gives Palpatine a motivation for the prequels besides simply "conquer the known galaxy." But, probably best, in my opinion, it gives you a reason that Vader would server Palpatine and the Dark Side. If transferring your soul is one of the hardest "spells" in Dark Side Magic and Vader has to learn it from Palpatine and also gain proficiency at using the Dark Side... well, it sure beats the "Join me and the Dark Side so that your wife can live forever. By the way, she's dead and you killed her, but joine me anyway." logic of Revenge of the Sith.