Opeth: swedish band that started as progressive death metal and over the years mutated to a more "traditional" prog rock in their last albums. The key element that gave Opeth the reputation of being the "best" metal band ever for so many people is the exquisite blend of brutal sound with extremely melodic/beautiful/melancholic passages. If you're not used to growling (singing to sound like a monster) I'd suggest to give it a try to a couple of their heaviest albums (Ghost Reveries, Blackwater Park, Deliverance), if you listen to them a couple of times, there is a fair chance it will click and then you'll appreciate how fantastic musicians they are. Three examples of those albums...
By the way, Steven Wilson produced Blackwater Park, Deliverance and Damnation. So, if he's involved, you know it will be pretty damn good.
If at the end of the day the heaviest aspect of Opeth it's not your cup of tea, you have plenty of other alternatives in their catalog. Albums like Heritage, Pale Communion Sorceress and In Cauda Venenum have no growls. Also, when they were at the peak of their most heaviest sound they released Damnation, which was a surprise cause it's a very mellow (with a sinister vibe) album that earned them a lot of applause.
Liquid Tension Experiment: You mentioned Dream Theater, well... in case you don't know, back in the day they had a sorta parallel project called Liquid Tension Experiment, it's basically Dream Theater without vocals, and with the legendary Tony Levin on bass instead of John Myung. The sound it's like Dream Theater on steroids which it's pretty bizarre considering DT sounds a lot like they already are on steroids xD
Riverside: These guys are from Poland. Progressive Metal but more melodic and less over the top, compared to Dream Theater. That allows them to make extremely catching tunes with kick ass guitar riffs (they also have very beautiful ballads).
By the way, Mariusz Duda, Riverside singer and bass player has a project called Lunatic Soul, it's progressive too, but with some electronic/ambient elements, it's pretty good IMO.
I also forgot to mention that Kevin Moore, the first Dream Theater keyboard player, started a project called Chroma Key after he left DT. It's a weird mix of progressive with avant-garde and a touch of electronic. Regardless of the genre, I love it and I applaud him for taking a totally different musical direction.
Haken: These guys are from England and also fall under the progressive metal umbrella. While they have some growls here and there, overall the vocals are clean and super melodic (the same can be said about the music, very catchy).
Animals as Leaders: This is a band formed by Tosin Abasi, one of the most impressive guitar players that have appeared in the last 15 years or so. The guy plays 8-string guitars so you can already imagine his sound is HEAVY. At the beginning the band consisted of himself only, so he chose to use a software for the drums which gives the first album a kinda electronic vibe that I personally love.
Plini: This is a young fella from Australia, one of best guitar players that I've had the pleasure to discover in the last years. He has a great amount of skills but also he never stops being melodic, he knows exactly when to press the gas pedal and shred without tiring the listener ear. His music is kinda hard to classify cause there's a progressive element in it, but at the same time is nothing like all the other examples that I've put here. Plini himself defines it as Happy Prog.
OSI: What the hell OSI means? Office of Strategic Influence, another project from former DT keyboard player Kevin Moore and Jim Matheos (Fates Warning). This is great prog rock with some electronic elements, very refreshing sound to me.
https://www.youtube.com/watch?v=9qd8-v4VWVc
https://www.youtube.com/watch?v=CufGvFqAR3c
https://www.youtube.com/watch?v=mRuSoU_Pw4o
Planet X: A fantastic progressive band founded by Derek Sherinian (another ex-DT keyboard player) and Virgil Donati, who is to me, pretty much the only drummer that deserves to be called "mutant".
https://www.youtube.com/watch?v=cQ3hUySljVY
This video is a proof of how amazing Donati is (oh yeah, Doug Shreeve is ridiculous too)...
https://www.youtube.com/watch?v=w4aEz_nr8p8
On the Virg: Another Donati project...
https://www.youtube.com/watch?v=zL_oMf3YvRk
Donati also has a very strong fusion background (on his own and collabs), sadly, since it's not a popular music, it's hard to find links to put here, anyways, some examples...
https://www.youtube.com/watch?v=pMHwzFOfNb8
https://www.youtube.com/watch?v=D6_fQIqLdls
Regards!