What kind of story do I want to see? Depends on what kind of game we're referring to. (I know, me getting complicated again. Who would've thought.)
Nintendo's talking about emphasizing non-linearity, so the best way to pull that off is most likely to have something of a Majora's Mask approach (or Metroid Prime if you prefer), where there's little actual story but a world of character development, subplots, relationships, and dilemmas throughout the land if you look for them. I'm not saying the story has to be dull or not improve, but it's hard to have one solid narrative when we're doing dungeons in whatever order we want. The enjoyability of the story then largely hinges on the quality of its segments we put together at our choosing. Are we angered that the daughter of the mayor was slain by the villain's personal attack on the village? Did we find love with a forlorn woman on an island? Are we coming to understand the villain more and more through our altruistic efforts? It's things like these that bring out the best of a loose narrative since it can't progress in one solid fashion. In a way, we need to be motivated apart from the story rather than because of it. Again, actual plot points and detail should still be deep and of the best fiber, but expecting the motivation to come from one solid narrative will probably lead to disappointment.
Should Zelda attempt another linear story (as I believe it should frequently), though, we're going to need a bit of everything stepped up. As JuicieJ said quite awhile ago, Skyward Sword did much to step up cinematics and character development, but it's a long way from home, and it's still sorely lacking in the detail department. Most of its story can be easily summarized in a few simple paragraphs without omitting any crucial details. What we need is a long, detailed, intricate narrative with plenty of CONSTANT character development and story reveals (like a JRPG), the expert cinematography and scenarios of a high-budget fantasy movie (like The Lord of the Rings), and realistic, varied, engaging character dialogue (like, say, Uncharted or Far Cry 3). Not every game combines all these things into one, so Zelda could actually be quite unique if it does this. It generally has good dialogue and engaging characters, but forgets to flesh out the story, add details, cover up plot holes, or pace out the game so it looks and feels as epic as it should. We need a significant overhaul.
As for smaller wishes for a linear Zelda story, I'd really like to see a Zelda game that doesn't rely on collecting a bunch of sacred objects or the "three dungeons, plot twist, three dungeons" formula. If only just once, I want a Zelda story that is constantly changing and evolving without falling back on tired tropes to move the plot forward, so that each dungeon comes with a very different task and ends with a different outcome, not just "go to this temple, collect the sacred item, open up the next dungeon area." That would be something new and refreshing.