Passau in Germany, near the Austrian border. It has the biggest cathedral organ in the world.
Linz, one of the larger cities in Austria, has a fine, large neobaroque organ where frequent concerts are given.
Melk in Austria, not far from Vienna, has a very impressive monastery (with several organs, but I don't know if they are special).
Esztergom, north of Budapest, has the cathedral were emperors were crowned. It's organ has been sampled for Hauptwerk. In and around Budapest several more organs have been sampled for Hauptwerk, see the websites of Inspired Acoustics and Augustine Virtual Organs.
But visit any church you can, be it in a village or a city. Many times you will be surprised by the architecture, and with a bit of luck and asking, someone is willing to assist you to the organ, or you'll even be able to play the organ directly.