I get this sense that the major differences can be summed up by saying, "In Morrowind, you were a small but somewhat important part of the world. In Oblivion and Skyrim, the entire world revolved around you."
I agree with this. In Morrowind, the only time you are deciding the fate of the world is at the end of the main quest, and even then it's implied that only people who worship the Tribunal and some member of the Blades will ever fully appreciate it while in Oblivion and Skyrim most quest lines end with you leaving a major mark on history. Additionally, in Morrowind, you are in the backwaters of the Morrowind province, which is usually considered a backwater region itself and the dialogue tends to reinforce that. In Oblivion, you almost never hear about the other provinces, though Skyrim was a bit better in that regard.
Anyway, the quest lines do have stories, like other people in this thread have said, but they usually aren't obvious. The Great Houses are about the politics of Morrowind and certain members of that House, the Fighters Guild is about corruption within the guild, the Thieves Guild is about restoring an old order of thieves and a shadow war with the rival Commona Tong, the Mages Guild is about what happened to the Dwemer and who the Dwemer were. The Imperial Cult quest line is sort of an oddball in that it's more your story. Not the story of the Nerevarine like the main quest, but just your character's story, and even then you aren't really important at all; it's just stuff that happened to a worshiper of the Nine. I haven't finished the Temple or Legion quests yet so I don't know much about them.
What's really nice is that a bunch of them tie into the main quest in some way. The Fighters Guild, Thieves Guild and House Hlaalu quests have to do with the Camonna Tong, which is connected to House Dagoth; The Mages Guild spends some time on the Battle of Red Mountain which is an important part of Nerevar's life; one of the Redoran councilors was killed by corprus beasts shortly before you arrive, etc.