As far as bands that had an obvious influence on Dream Theater's sound, I would definitely recommend Deep Purple, Queen, Yes, Pink Floyd, Rush, Rainbow, Iron Maiden, and Metallica.
As far as other 80's metal bands who, like Dream Theater, helped lay the foundation for progressive metal as a sub-genre in it's own right, you should check out Queensryche, Fates Warning, Crimson Glory, Watchtower and Savatage (for more recent materiel by Savatage members, check out Trans-Siberian Orchestra, Jon Oliva's Pain and Circle II Circle)
As far as bands that have come after that were influenced by DT, Symphony X is far and away the best, IMO. Most of the others, like Circus Maximus, while good, do have that sort of "clone" sound to them. So for more recent prog-metal bands, I would suggest going for bands that share much of the same influences as Dream Theater, rather then being directly influenced by DT themselves. Balance Of Power, Kamelot and Ayreon (and the various other projects of Arjen Anthony Lucassen) would be good places to start.