If I was arrested for DUI but never read my rights, can I get my case thrown out?
It depends. The Miranda warning need only be read to a person is if they are under arrest and then questioned by the police. The exception being that questioning incidental to the booking process is allowed without the warning having been given e.g. name, address, birth date, etc.. Additionally, the charge would not automatically be thrown out. Even if you should have been read your rights and were not, the statements made by you would not be admissable in court, however other evidence might be strong enough to obtain a conviction. In other words, you could still be found guilty of the charge.
At this point, you should consult with a DUI attorney directly about your case. They can best further advise you.