PREMISE: The OSR is organic.
There are, I think, three aspects to an organic quality in old-school RPG'ing: Achievement, Neoteny, and Serendipity.
What do I mean by "organic"? Well, it's like growing tomatoes in your garden.
ACHIEVEMENT: You've planted this tiny seed in the ground and it's grown into a vine bursting with tomatoes. Tomatoes YOU have grown. No one else. It's a sense of accomplishment, and, of course, you have to regale friends and co-workers with stories of how delicious those tomatoes are, all the more for the fact that you grew them. And maybe give some away. Delicious tomatoes! Try 'em!
Achievement comes from both writing stuff (which is partly why OSR blogs are full of "LOOKIE WHAT I DID!") on the DM side, and in nurturing a character on the player's side. OD&D characters can die like flies, which is no biggie since you just take forty-five seconds and roll up "a relative" and have him wander out of the forest into the midst of the party and go from there. But when you do get that fragile Veteran or Medium or Acolyte up to Second Level, there's a sense of having accomplished something.
It's like Zak pointed out awhile back:
That "Riddles In The Dark" effect is why I like the Old School approach to plot and character and epicness and awesomeness.
Which is: you start with none of those things. You start by sucking. You start by sneaking. You start with one hit point. You start with no plot. You start anonymous and meaningless and arbitrary. You have three torches and a short sword and whatever armor you can afford and no feats or skills in a dot on a hexmap hitting another dot on a hexmap.
From the DM side, think about all the stuff that grew (organically!) out of OD&D and those three lil' books - not only the D&D gaming worlds like Greyhawk and Blackmoor and Arduin but entire other universes like Tekumel. Out of something that looks like typewritten notes in the back of someone's folder! That's achievement.
NEOTENY: Playfulness, creativity, innovation. You design your own tomato garden; nobody else has one exactly the same, even if you're working out of a book from the library. That plot is unique and what you do with it is your choice. Just tomatoes? Or maybe lettuce as well? Stick some peas in there somewhere? Nobody's going to come along and slap your wrist and say "NO THAT IS NOT THE RIGHT WAY".
OD&D was like that. Shared worlds? Hell, there were barely shared rules. Every DM ran into something he had to rule on, and the accumulation of individual house-rulings made the D&D campaign over here entirely different than the campaign over there. Over here robots and machine guns, over there pure fantasy. Over here, PCs kicking Odin and Loki's asses; over there, rugged Conan-esque swords & sorcery. Even the rules themselves mutated; use the "to hit" tables as written or come up with your own alternative? Do I want to be limited to +5 as the best enchantment on armor or weapons? Here's my "spell points" system so I can get rid of that Vancian crap. There's a detailed set of magic research rules (from The Dragon #5, reprinted in Best Of The Dragon #1) which I still marvel over, particularly the "The Sorcerer's Memorial Enchanting In Distress Rules". I have no idea if I'd use them in a campaign; they look severely broken. But I'm sure someone did. And they weren't "broken" for his world.
SERENDIPITY: Oh boy. Oh boy, oh boy. There are quite a few tomatoes on that vine. Didn't expect that. Can 'em, cook 'em, freeze 'em, hand them out to people on the street. "HERE, HAVE SOME DAMN TOMATOES!" The family starts complaining about the proliferation of gazpacho and marinara sauce and tomato salad. I just freakin' planted some damn tomato seeds. Who knew?
On the gaming side, us OSR types sure do love our random tables. Whee! Tables! Monster encounter tables, monster building tables, "stuff in a wizard's workshop" tables, random NPC tables, tables tables tables. Give me a bunch of tables and a couple appropriate dice and I'm a heppy, heppy "kat".
So are we really a bunch of morons whose only defense for a Vampire with a cow's head and three red dragons in a 10x10 foot room is "I ROLLED IT ON THE TABLE!"? No, please.
The tables are there for inspiration, which is why most of them shouldn't be used at a game, but before. (Which is a problem I have, as evidently others do, with the was-going-to-be-great DCC RPG. Although, you could just roll up a list of random results before the game and consult it when a spell goes wrong, I guess.) The dice are there to show you something you might not have thought of on your own, due to personal bias, gaps in experience or whatever. It's like opening a lil' Christmas present - what will I get?
There's also the quality of the unknown to Serendipity - the thing of "Hey, I didn't know it could do that!" q.v. the first time a new player runs into a Wand Of Wonder or a Bag of Beans or and Amulet of The Planes - or, better yet, one of those sweet artifacts from Eldritch Wizardry. Whee, a +5 Holy Defender sword! Only, why does everyone else's magic items keep getting drained...? And why are the hirelings so pissed at me?
OD&D was highly organic on all three counts. 3d6, in order, your M-U starts out with 1d4 hit points and a rusty dagger, 1st level Clerics get no freakin' spell. But keep on with it and pretty soon the Sorcerer is throwing Fireballs and the Cleric's doing Raise Dead. Achievement. Neoteny? Hell, you're barely given a set of rules, for Kord's sake! DM or player, you're going to have to innovate to get this thing working properly. Serendipity? What the hell can I do with this character that I rolled up? What's that magic sword that I found do, exactly? Hell, one of the very first articles about D&D, 'way back in Strategic Review #1, was Gygax's solo random dungeon generator. I also think of Paul Montgomery Crabaugh's "random monster" tables from Dragon #10 - which, he points out, are presented specifically to keep a sense of mystery about monsters for players who know 'way too much about the rules.
First Edition AD&D was organic - it kept the limits on beginning characters, but started the ossification of the rules, ideally in the name of convention games but possibly because someone at TSR - Gygax, whoever, - thought they needed a better rein on what was being called "D&D". But Serendipity - yowza! Just take a look at the back of the Dungeon Master's Guide - random tables out the wazhoo, including a reprinting of the solo dungeon generator. And they gave you this shit in a core book! Man!
Second Edition I don't have a lot of knowledge about, but it seemed to be less on the organic quality - you had a lot more options, but the rules seem to have ossified further. This was also the golden age of shared settings - Dark Sun, Planescape, Forgotten Realms, Dragonlance, Mystara, pseudo-Greyhawk, Ravenloft. Which were great and all, but someone else was doing the universe-creating. And (as far as I know) all the cool tables were dropped from the back of the DMG.
Third Edition - even less organic, what with forgetting about recommendations for building your own world as opposed to gaming in one of their pre-made ones like XCrawl, Eberron, or Forgotten Realms V2.0. Achievement, meh - creating a character became a lot more complex all of a sudden, meaning instakill poisons and energy drains got nerfed. And the magic item list lost its cool artifacts in favor of "official" artifacts, with a list of "official" powers and drawbacks. And no more random tables in the back of the DMG.
Renfest D&D - hell, it's nigh impossible to die because that has been decided as "Not Fun", everyone is awesome from the start so Achievement becomes just BECOMING EVEN MORE AWESOME, the magic items are in the Player's Handbook for crissake, we've even disposed of wandering monsters, and there aren't any more secrets or surprises because that might inhibit taking full tactical advantage of one's situation. Lists of "magic items I want to get next level"! JESUS! You might still have the Neoteny, though that's pretty much it. And with an "official" character generator, that's that much less opportunity to write your own rules; give characters healing potions instead of healing surges, limit powers, that kind of thing.
So there you go. "That is the theory that I have, and which is mine, and what it is too."