Wow, some great responses here---I'm going through the last two pages.
I'm definitely in the camp that character backstory makes a difference. A character with a sense of "place" in the game world has a lot more opportunities for "narrative"-inspired play.
I'll try and chime in with my thoughts on the question.
My experience is that any drive for your defined 'narratavism' play has to come from the players. It doesn't come from GM-ing techniques, or game prep, or relationship maps - all those are enablers. The only driver is player desire to play that game. If it's not there, you're fighting against the tide and you'll lose.
Assuming the players want to explore moral questions, face ethical dilemmas, express judgements about the world through their characters I think there are some product neutral things you can do. Collective world-building is very good. Letting everyone build the entire setting gives everyone that sense of place, and an implicit knowledge about the world, which allows them to be proactive.
Coupled with collective character building it allows everyone to establish straight away their goals, interests, relationships and conflicts.
All that information is vital once you start GM-ing this kind of set-up. Many of the more 'narrativist' games write this in either procedurally (FATE) or as 'How to run this game' (Apocalypse World, Burning Wheel).
In my opinion, the biggest block D&D puts on 'narrativism' is that it assumes that every character's goal is to level up. Fundamentally, D&D is about levelling. In AD&D levelling was the benchmark of 'skilled play'. Skilled play was not portaying a man seeking revenge, and eventually redemption, after he failed to protect his sister from that troll. It was making it to Level 2 - and that hasn't changed much.
If I were inclined to run a 'narrative' expirement with D&D I would make two simple changes.
a) Instead of XP for defeating monsters, disarming traps, looting treasure (see how those dictate to players what constitutes success?) I would have each player write 3 goals - one short term (can be done this session) one medium term (can be done in 3 or 4 sessions) and one long-term (can be done in 10 sessions). I'd give XP for, and only for, working towards and completing goals. Basically, the Burning Wheel advancement system simplified for D&D. Of course, writing goals requires already knowing something of the situation - hence the collective world-building.
b) Change the pass/fail mechanic. Player states intent. GM calls for a roll. Any roll that beats the target by, say, 3 and the player gets what they wanted. Any roll +/-3 and the player gets some of what they wanted but the GM can introduce a complication, dilemma or hard choice. Miss by more than 3 and you fail. So, instead of needing a 13 to grab the woman as she teeters on the balcony you need 10-16, but you might get offered the choice of grabbing the woman but dropping and smashing your potion of invisibility or not grabbing her. Hard choice. Which is worth more to your character right now, in this situation? You've put an ethical dilemma into play from a simple situation and which the character gets to use to help define their character. Over a period of time this allows vivid characterisation. This method comes from Apocalypse World (and so is in DungeonWorld too).
I can only repeat, the players have to want this game. If you've got the kind of players who mistakenly shout 'railroad' the moment you offer them a real decision (The girl or the potion?) then it's the wrong game for that group.
Hope that helps, or at least provokes some new ideas.