First, check what you like in Dream Theater
Dream Theater mixes the Progressive aspect from (mainly) Rush, Deep Purple and similar bands into the Heavy Metal aspect from bands like Metallica, Black Sabbath and Iron Maiden as their influences by the time. They also have a thing for fast playing (although not being the only thing they do, before someone downvotes for that), which is a predominant characteristic on Power Metal bands.
Either way, really similar bands
- Haken is a really, really similar band to Dream Theater, independent of your choices above. From my contact with the fanbase, if you are a fan of one, you're most likely going to enjoy the other.
- A.C.T. also has many musical similarities, especially in riffs. Circus Pandemonium is a good album to start, in the sense it shares many progressive metal concepts from DT.
- Tool, while not that similar, is also often liked by Dream Theater fans.
- Symphony X, while being more focused in the Power Metal than the Progressive Metal, is also commonly associated with Dream Theater.
- Liquid Tension Experiment is essentially Instrumental Dream Theater.
- X Japan is often called "Japanese Dream Theater" - note that japanese jazz music is usually even more dissonant than western's, and that is true for their prog as well.
The Prog aspect
If you are into the prog aspect, you will likely be enjoying old prog rock bands, such as Rush, Yes, Gentle Giant, E.L.P., King Crimson, Jethro Tull, Camel, Pink Floyd, Deep Purple (ok, DP might not be considered prog, not discussing it) and similars.
If you are interested in recent bands, you should probably follow the works of Steven Wilson (Porcupine Tree), Neal Morse (Spock's Beard, Flying Colors), Mike Portnoy (Dream Theater) and similars.
Other than that, really, progressive metal is a genre with too many bands. A few with renown (presented, for example, in the Essential Modern Progressive Rock Albums by ProgReport) are Queensryche, Marillion, Spock's Beard, The Flower Kings, Devin Townsend, Opeth, Transatlantic, Pain of Salvation, Porcupine Tree, Ayreon, IQ, Riverside, Frost*, Karnivool, Between the Buried and Me, Big Big Train, The Dear Hunter, Rishloo, Leprous and Sound of Contact.
A few bands also have singular albums that are good, but mostly their other works are not well received. The Mars Volta has De-Loused in the Comatorium, for example. Native Construct has their debut and only album until now - Quiet World.
Either way, it's a big and broad genre. These bands should give you a start, after you know better what you are enjoying, you can pinpoint your searches better.
As a special note, since you are asking about DT, it's interesting to look for bands Portnoy has participated. In particular, his interactions with Neal Morse usually end up in great progressive projects. Examples are the already mentioned Transatlantic, Flying Colors and The Neal Morse Band. Sons of Apollo is his most recent project.
The Heavy Metal aspect
Well, this one is even broader, but AFAIK, less common for recent bands. A fair share of the DT's fanbase is interested in plain heavy metal, such as Iron Maiden and Black Sabbath. If that's your case, simply look for more heavy metal, not for more progressive metal.
The Power Metal aspect
If you are instead interested in the fast playing and high tones, you can go for Power Metal. Here, you can go for Gamma Ray, Blind Guardian, Helloween and Iced Earth as things that a Dream Theater fan might enjoy. This is more likely to be true if you enjoyed When Dream and Day Unite.
If you enjoyed the concept of a really long Opera Rock album (a.k.a. you liked The Astonishing), Ayreon gets a stronger recommendation, as well as the a little bit underground band Gandalf's Fist, known for having 3 hours albums.