Just say no.. Tell your health provider no. No details need be provided to anyone else except that you are not an acceptable donor for the potential recipient. Period!
Tell your family no. If you are certain that you do not want to donate organs (=/or tissues) after your death, discuss this wish with your family. You should also place this wish in a statement on your healthcare proxy. It would also be easy to write out a statement, have a witness sign it and carry it in your wallet. Organ donation teams simply need to understand your wishes and they will always be respected.