- Everyone gets a thing to do, even if it is just them choosing to assist another player with their area of expertise
- Mechanics need to be involved in someway. This is the Characters exploring, not the players.
- Mechanics can't be so involved as to bog things down.
With those in mind, 5e gives a number of options. Some of these options translate to other games cleanly. Some do not.
Skill Challenges in X Minute Increments
The first, and probably most universal, is to simply go around the table and have every Player say what their character is focusing on for the next X-minute interval where X could be 10 minutes of checking a dungeon, 60 minutes of spelunking a cave system, or even a day of travel or however you like. Each player chooses a skill to represent that task - it is up to you if duplicate skills are allowed, encouraged, or penalized in some way. They then make a single check, and based on those results you narrate to the PCs what they come across in that time period and how it relates to what they did.
For example, the person keeping an ear out for approaching threats maybe gets a description of some of the strange sounds and smells they perceive while staying alert for monsters. The person who is checking for tracks can get some information about the construction and flooring, along with what types of traps and pitfalls they're noticing along the way. The person making a map can start to get a sense for the layout and structure, or even the purpose behind things.
Skill Challenges With Passive Scores
This is more or less the above, only instead of dice rolling you just use their passive score. A passive score is a 5e mechanic where you take the result as if the player rolled a 10 on their D20. A 10 the (rounded down) average result for a roll of a D20, so in other games you could just use the 'average' roll for each player.
You then compare these results to the challenges and treat it like the skill challenge above. Characters not focusing on alertness type skills (i.e. Perception) get disadvantage on it due to being pre-occupied (in 5e disadvantage is a -5 penalty to the passive score) but can still notice other things.
Since a Passive Score uses the average it makes sense to work for a prolonged period of checking.
Only The Important Parts Need Rolls
In any given dungeon there are going to be important obstacles, and obstacles that are just there as part of the defense. A group of random wandering monsters is less important than the monster guarding the boss key. The locked/trapped door leading to storage closet 4c is less important than the door to the secret passage that bypasses half the traps. With this method you don't roll for those things, you just touch on them quickly and keep moving.The dice only come out for the important obstacles. This does mean unimportant traps/encounters won't wear the group down, force rests, or perhaps even turn them back, but it also means the game doesn't get bogged down in minutia for things where it isn't warranted. Your players will catch on quick that you're only asking for rolls by important doors, and that alone can help raise the tension because those doors matter.
My Preference: Skill Challenges With Consequences
My personal preference right now is skill challenges, but with consequences. What I mean by this is two fold: first, if players try to double up on a skill (assisting is fine, but two people using the same skill) the DC is higher for the second person. This is to encourage breadth of skill usage for the exploration being done, which in turn increases the amount of information the PCs can glean from the exploration. Second, is that low rolls will trigger some kind of encounter which can also be used to further the story of what is being explored.
These encounters aren't all necessarily combat or dangerous, but they are obstacles that pop up and need more player time on account of the skill check. The door that could have just been opened, now must be resolved as a puzzle. The secret passage way that could have been spot, means the PCs notice they're going in circles and have to figure out if they're going to backtrack or keep going in hopes it is just a trick.
My problem with this is that while it works well enough for navigating a dungeon, I'm not sure how well it works for exploring a frontier and that's still a challenge. But at least it is a start.