The question arises because some commenters at File770 brought up that minis were unnecessary, and one pointed out that to his kids “Old School” meant “that the game not be bogged down in setting up miniatures/toy soldiers or taking long breaks in the action to consult the rules”.
I’m all for games that don’t require taking long breaks to consult the rules, but I’m also wary of simplifying play by discarding them altogether.
So, can you play Dungeons & Dragons without minis? Of course you can, but it would be akin to playing Monopoly without dice; players would choose which space they landed on to buy whichever property they wanted unless the banker capriciously says otherwise.
I’ve played several RPGs where minis weren’t used for encounters. I’m not saying it can’t be done. But in the case of Dungeons & Dragons, the game’s rules assume that you’re using miniature. You have to toss out a significant chunk of the game’s mechanics to get away with not using them, because suddenly movement rates, weapon and spell ranges and areas of effect become completely arbitrary, left to the DM to decide whether or not a person is where they say they are or doing what they say they’re doing. In a way, this is incredibly unfair to players, each of whom may have a different mental picture of what’s going on in a given encounter. When you have minis out, everyone knows exactly where they are in relation to each other, their opponents and physical obstacles in the environment.
I’m not saying that people who aren’t using miniatures are bad gamers or their games are bad or unfun – I’ve had great fun games in that didn’t use minis – but I will say that they’re playing it wrong. Because when you’re not playing by following the rules, how else can you describe it?
Seriously, though, if you’re running a game or playing in a game, just try it sometime. You’ll be surprised at how much more exciting and involved your encounters can be when everyone shares the same view of the combat situation.
Cirsova 
There’s using minis, and then there’s using minis.
In the 5e campaign, we go strictly by the book. Five foot grid, full color maps, rigorous attention to the rules. Lots of square counting, strategic talk, and that sort of thing. Combats take forever. We’re lucky to get in 2 during a 4 hour session.
My OSR game uses them, but not strictly by the book. We throw down a quick sketch of the battlefield showing walls, obstacles, and other important features, then place minis to show relative locations.
Here is a bad picture of it: http://4.bp.blogspot.com/-5LgPI5N2wvk/UKXvJSq6P0I/AAAAAAAABS0/SnVLhHS6m7o/s320/GEDC2366.JPG
We don’t use strict movement rates or missile ranges, though. For that we rely on DM fiat. Keeps everyone on the same page for positioning, but keeps play from bogging down in analysis paralysis. We can get through 4-5 combats in a four hour session.
I ran a B/X game with minis awhile back at a library program where we’d only ever have an hour and a half to play, but we’d often get 2-3 encounters in per session.
In the DCC game we’ve been playing, the only part that has really bogged us down has been spellcasting; movement rates and range hasn’t been an issue (we’ve just made sure everyone has their move and ranges written on their character sheets)
Now, where it gets sticky is if you try (which I have not) going with Holmes’ proposed 3.33333333′ squares to allow for combat 3-abreast in hallways.
But I do think that having a visual representation of encounters, even if it’s just with dice or coins, is important to fairly handling encounters within the rules of the game.
I like those trees.
They paint up even nicer. I got ’em at the local craft store (Micahel’s/Ben Franklin), but you can order them on Amazon: http://www.amazon.com/Safari-Ltd-684304-Trees-TOOB/dp/B001Q0Y5GS
I love using minis and probably would not have got back into RPGs without them. That said I don’t think any of the editions before 3rd really required representing things with tokens and grids. AD&D pretty explicitly assumes you won’t use them. Look at the rules about randomly assigning targets — even in melee.
But like I said, I prefer using minis. They add something that my group enjoys at any rate.
The fact that everything’s movement, range and AOE is in inches suggests to me otherwise. I mean, ranged weapons have accuracy bands. It can be done, but for B/X and 1e, it means discarding half of the combat engine in favor of just rolling to attack whatever you feel like and the DM says whether you do or not.