While it sounds like a contradiction in terms, to propose a "diceless" d20 rules system, that is in fact exactly what I'm proposing.
Others have developed inventive and notable alternatives to d20 that either don't use dice or de-emphasize them in favor of narration (Amber Diceless, Mortal Coil, Wushu). Other systems like Dirty Secrets, Primetime Adventures and Dogs in the Vineyard not only emphasize story telling, they are essentially story telling games.
My attempt with d20 Narrative is a bit different. I'm not trying to replace the d20 rules system or the d20 "character-centric" feel. There is still a game master who primarily controls the story, setting and in most respects, adjudicates, and there are still players who primarily control their own character. There's a great deal of history to the d20 mechanics as well as ubiquitous familiarity with d20 style RPGs. D20 Narrative attempts to leverage that while taking some ideas from the "story games" to promote narrative game play. While in some respects, this is a clear departure from the dice mechanics of d20, this could be viewed more as a d20 variant, rather than an alternate rules system.
While d20 is a well rounded gaming system, too often the story is hijacked at conflict resolution, like "Roll initiative." At that point the story takes a back seat to the rules system used to resolve the conflict. What might have been an immersive story at that point switches to a series of dice rolls, modifiers, and strict turn-based activity. While the players choose their own actions, their success or failure is determined by their statistics coupled with random dice rolls. While many game masters and players make a conscious effort to maintain the story by narrating their actions, the narration has no impact on their success and often is left out entirely.
In published adventures, the adventure usually starts off with a series of Gather Information rolls to feed the party background information. While this might serve as a useful tool in some cases, the "randomness" means that some parties are going to have considerably more or less information to start the adventure with, notwithstanding the immediate use of the dice, setting the tone; lending to more "roll play."
D20 Narrative takes a different approach. By applying the concept behind the existing checks without rolls (i.e. "Take 10" and "Take 20"), this system resolves conflict based on narration. In the standard rules, checks without rolls may only be used when not threatened or distracted, or when the character has time. Removing these restrictions, since all resolution in this system is determined without rolls, you are left with the quality of your narration (not "quantity") determining success.
For example, in a typical d20 dungeon setting, the game master describes a room with a doorway the elven rogue finds suspicious. Normally the player would call out a "search" check and make a roll or she might specify she's taking her time for a "Take 20." That modified roll is then compared against the search DC of any "hidden" items on the door, trap, secret opening mechanism, etc. If the player succeeds against a particular search DC, the game master rattles off the results.
In d20 Narrative, the game master describes the room in similar fashion, however, the player with the nosy elf must now "narrate a search check." Unlike a typical d20 game, no dice are rolled and there is no break in the story. In order to attempt a search the player must narrate what it is the player is doing. For example, "those carvings in the door are giving me a bad feeling. I carefully approach the door, examining the floor and walls around it; focusing on the hinges and the carvings." At this point, without saying anything the game master looks behind his screen at the stats, both the player character's and the doors (if any) and replies, "While the carvings are unnerving, the door appears to be just that." This might be considered a standard "Take 10", though it was never mentioned and the story continued unabated. If the door indeed had something secret or hidden specifically regarding the hinges or carvings, the GM may add a circumstance modifier to the "roll." However, all these mechanics are handled silently by the GM and all rapport between the GM and the players is strictly narration. The player attempting the task may add circumstance modifiers with her narration, "I run my fingers along the door's handle, to see if it's as smooth at it appears." Other players may "Aid Another" for additional circumstance modifiers ("I hold my torch close to to shed more light where she's looking.") Again, the mechanics are implied and the only verbalizing is the narration.
At this point, the core d20 Narrative mechanic is checks without rolls, with the caveat that all conversation is narration, in some cases in third person, but primarily first person "in character" and the GM adds (or subracts) modifiers based on the narration. To take this a step further, and to give more control to the players, each player has a "vote." At any point during conflict resolution, before the GM adjudicates the results, all players can indicate with a thumbs up an additional bonus for any narration that sounds particularly interesting or adds significantly to the overall feel of the game. In a typical game, narration that sounds implausible, or where success might negatively impact "suspension of disbelief," players are encouraged to give a thumb down or thumb neutral (neither adding or subtracting from the attempt). This should vary widely depending on the game type. Superhero games, space opera or wild west settings might actually encourage the more fantastical, while in games of hard science fiction, historically based settings or any setting where the characters are essentially normal, players would tend to vote down these types of actions. It all depends on the "feel" the group wants to instill. Ultimately, the Game Master has control as he might increase the Difficulty Class depending on what the player is trying to accomplish with her narration, however, GMs are encouraged to stay in touch with the feel of the party and adjust accordingly.
There is of course a degree of "cheating" that might go on. For example, the entire party may indicate thumbs up after a player simply narrates "I attempt to pick the lock" because they are motivated to get behind the locked door, but players and GMs alike should be engaged to a point where the narration improves over time and thus the group continually raises the bar for a thumbs up. Like most things, you'll get out of it what you put into it. Having said that, a high degree of familiarity with the existing d20 rules set will allow players and GMs alike to narrate more naturally, without having to extract the implicit mechanic and dig through rules books to determine the chance for success. When in doubt, make a silent judgement call in-game and research it outside of the game, so the story continues uninterrupted. Breaking the narration to discuss rules is highly discouraged and at odds with what d20 Narrative attempts to add to your game.
When it comes to combat, the d20 Narrative should allow the story to flow as well as give the players more control over the outcomes of their battles. For players unfamiliar with all the minutia of the d20 combat rules, some technical discussion may be necessary during combat resolution. However, with seasoned d20 players, combat should remain entirely narrative, interjecting "attacks of opportunity," feat usage, special attacks and the like strictly with narration.
Strict turn-based initiative, move, standard action, full-round action can be adhered to or you might want to "loosen" the rules here in favor of narration. Strictly taking 10 for initiative is going to favor those with high initiative modifiers and make turns occur more predictably. That may be desirable or not. Alternative modifiers to intitiative might include what was last narrated by each player, marching order, proximity to the threat, etc. Alternatively, if you like the random aspect of the dice, but you want to try the d20 Narrative variant, you might want to have some pre-rolled random numbers or a spread sheet so that some things may remain random without interruption of the dice rolls. However, the important item to stress is that the player should have a considerable degree of control over her success with the quality of her narration.
There are some aspects of the game you may wish to remain random, so the degree to which you use this variant is up to you. However, entirely removing dice and restricting all in-game conversation to narrative should give your game a truly immersive feel and the quality of the role play should improve with each conflict resolution. Ideally, the gaming group will raise the bar with each conflict resolution as the quality of the narration increases.