If the police don't read me my rights, can I get my case dismissed?
Not necessarily. The Miranda warning provides notice of the right against self-incrimination and protects a person from having their statements used against them if the statements are the product of a custodial interrogation.
If a court were to rule that such statements were inadmissible in court, that ruling would obviously have some impact on how the case would be resolved. Ultimately, the State would have to pursue the charge based upon other evidence. Unfortunately, the State may have other evidence independent of the inadmissible statement. If you have concerns about statements given to the police, please discuss the events with your attorney.