Describing the Current Situation
It's pretty clear to any casual obsever that 4e D&D is very, very, very focused on the individual encounter.
Three quick pieces of evidence, for those unconvinced: Encounter Powers. Short Rests. Milestones. All of these mechanics hinge on the nature of the encounter as the basic action point in the game.
Three supporting pieces of evidence, which are poor in 4e in part because they don't focus on the encounter: Rituals. Skill challenges. Noncombat powers.
Ghosts of Dungeons Past
Of course, things weren't always like this. Older D&D was much more focused on the effect of several encounters in a row, over the long-term exploration of the dungeon. This made things like the Rust Monster make sense: the creature wasn't an encounter in and of itself, it was part of a bigger dungeon, and its ability to dissolve your gear (and inability to do much else) was much more a part of the dungeon's overall effort to repel invaders than it was a specific effort to kill PC's in individual scenes. It was a long, strategic game, rather than a brief, tactical one.
Combining the Former with the Latter
Part of what 4e has going for it is very strong mathematical balance, that basically resets after every encounter (well, except for daily powers, but lets set those aside for the moment): you begin every encounter at full HP and with your full assortment of abilities.
In order to focus this more on the adventure, we need to change it so that at the beginning of every adventure, the party is at full HP with their full assortment of abilities.
Because this is kind of a new thing, people have their own ideas about what "adventure" means to them. That's all fine and dandy, but to talk about this thing in a coherent way, we need a shared definition.
Mechanically, the "adventure" is what happens between the party recovering their resources. Once the party can "restore back to full," the adventure is over, and the next adventure begins.
This parallels 4e (where you gain everything back at the end of the encounter), and also gives us a basis for making adventure-based resources and adventure-based challenges.
Like we've seen in balancing Vancian casting, we can get a good count of how many things a typical character does over the course of a 4e day or a 4e combat. Killing monsters can translate directly into other parts of the Three Pillars, too. Given that, we have an overall sense of how much a party in 4e is expected to do before it takes a daily rest and recovers all of it's resources.
It's possible just to dub that "the Adventure" and be done with it, if you'd like. In this case, if you use XP rules resembling 4e, you have about three "adventures" per level, each adventure consisting of three encounters (one per pillar), each encounter consisting of one challenge per PC. In actual play, this translates to something like one "adventure" per game session, with a level happening once each month. Which is a very comfortable rate.
You might, in this case, want to change how long it takes to rest, though it's not strictly necessary. If you do, you'll find a pace very similar to what existed in early D&D editions, with a given dungeon delve taking several days, and a recovery period that takes significantly longer.
If your adventures tend to be longer-term, multi-session events, you might want to dub the period between full rests the "Quest." The "Adventure," then, becomes three Quests long, and, when you complete the Adventure, you gain a level. This is mostly just a terminology jag, and it works well for games that meet regularly. A game that only meets for a single session might be described as having a "one-Quest Adventure," which doesn't result in a level-up, necessarily.
In this model, you could also replicate the old-school need for more encounters to gain each level. If Level 2 requires only 1 quest to achieve, you could have level 3 requiring 2, and level 4 requiring 3, and so on, until level 20 requires 19 quests to achieve.
This has bang-on effects for pacing that aren't necessarily great, so assuming 3 quests per adventure (or level) is a good baseline, but it's entirely flexible, since recharges are based on the Quest.
Recharging the Adventure
In D&D, deciding when to rest and recover is mostly a PC decision: they decide to camp for the night, or to retreat back to town. As discussed in the Vancian Magic musing, this can futz with characters whose powers are limited by frequency: they can undertake the adventure, quit halfway though to recharge, and come back to complete the other half at full power. It's more than just wizards who do this: if Assassins get surprise every combat, their Death Attacks are more powerful than normal, too.
There's a few ways to envisage this as I describe under What Recharging Looks Like, but the upthrust is that the adventure replenishes all its threats when the party replenishes all their resources.
So, the party can retreat whenever it wants, but when they do, they essentially re-start whatever adventure they've been undertaking.
A DM may want to change the circumstances a bit more drastically, and that's fine, too, but the important point is basically to require the adventure to be completed all at once, not in fits and starts.
Awards: Items, Pieces and Points
Because the game is based on the adventure, this is the container of the reward. Rather than getting treasure and XP after each encounter, the party gets treasure at the end of the adventure. This also encourages the party to complete the thing all at once: if they retreat, they're going to spend gold, which lessens the reward they might get from the adventure.
You could also introduce "secret" ways to get some awards (such as GP or magic items) early, rewarding the party for smart or lucky play. As long as the carrot at the end remains the juiciest and freshest of the carrots, having a few along the way isn't necessarily a problem.