So Mythic Dungeons are actually not all that difficult - they're beefed-up heroics, but they don't really expect that much more power from you, and mechanically they're the same.
However, there are two dungeons that are available as Mythic-only, and require you to do a lot of quests for the Nightfallen in Suramar - you need to be two-thirds of the way through Honored to get the pre-quests, and then you need to do a series of group quests (that I did solo because I did not realize they were group quests at the time - it was not easy.)
Frankly I think this attunement is a little extreme - it's not terrible on my main, but having to do this on any character who wants to go in there? (And given that there's an engineering quest that sends you there, I really have to assume that that quest was designed when there was going to be a normal/heroic mode of these dungeons and that there was no attunement.)
Both dungeons are unconventional, but in different ways.
I did Court of Stars first, so I'll start there.
There are only three bosses. The first boss is relatively conventional - you clear trash up to him and then pull him. However, there are some beacons that guardsmen can use to call adds, but you can deactivate these. Additionally, if you have an alchemist in your group, you can poison the potion that he drinks in the last portion of the fight - thus instantly killing him instead of dealing with an enrage phase.
The second part of the dungeon requires you to pick off demonic guards to draw off one of the three adds the boss has - picking these guards off and then the adds will make the actual boss fight significantly easier. Throughout this phase there are many things that people with particular professions can do to buff the party - though it was clearly bugged, as I was unable to interact with any of the little mechanical cleaning robots but I was able to activate some soothing spores from a plant with my nonexistent herbalism skills.
After the second boss is down, your contact will give you an improved disguise and you can enter a party. There you can talk to guests who will talk about rumors about Elisande's spy there. In fact, as a Protection Paladin, Truthguard will actually light up when you're near the spy, but we solved the logic puzzle before I could use that method. Following the spy after accusing him/her will reveal that it's actually a dreadlord. Kill it and you'll get access to the room with the final boss.
Arcway is in many ways a more conventional dungeon, but the main trick is that different doors will be locked each time you run it.
You'll want to make your way through the dungeon until you've killed all the bosses except the final one. You'll then make your way to the center and fight the final boss. At 50%, he'll teleport the party to the room of a random other boss, and you'll have two minutes to get back to him or he'll insta-kill you.
Along the way, there will be arcane elementals that will stun party members - you want to kill them quick and then keep moving.
In a way I can see why these were made Mythic-only - doing this in a truly random LFG group might be a real pain. But the attunement, for me at least, means that asking members of my guild to run it will probably mean waiting a long time.
I am really eager to hear details about the Return to Karazhan dungeon. That's something I'm hoping to run frequently with the guild.