|Even "D&D for Dummies" says so (via Google Books)
This tradeoff shines brightest when the DM applies old-school logic and throws in monsters that can't be defeated in a toe-to-toe combat, but can be run from. Each armor-wearer has to decide whether their armor makes them half as likely to be hit by low-level grunts, or lets them get away from slow and overpowering monsters.
The weird thing is that in D&D up to 3.0, plate mail is really not that expensive compared to the tons of treasure you are required to harvest to level up (xp from monsters being stingy). So cost doesn't figure much in the tradeoff - especially given that armor is a common form of loot. In my campaign, armor is expensive and monsters and carousing count for more, so treasure amounts can be moderate at early levels; character typically get access to medium armor around level 2 and heavy around level 3.
The other weird thing is that as you get magic armor, the tradeoff disappears - it gives both greater protection and mobility. In my campaign, magic items are rare and the standard improved armor comes in either dwarven steel (+1 to armor class) or elven steel (+1 mobility class), where each bonus is valuable separately.
But hold on! Isn't the mobility-protection tradeoff overhyped when you look at actual medieval armor?
Plate armor wasn't all that restrictive of movement.
Armor didn't have to be expensive.
Wearing armor slows speed only through increasing fatigue.
And leather armor affording the same protection as metal, although lighter, would restrict movement in the same way, because to be effective at all against weapon it had to be thick, or treated through boiling to become a hardened material.
Well, the sovereign answer to all of this is that gaming combat doesn't have to be realistic - in fact, should include any and all misconceptions that are crucial to a fictional genre.
But here's the more satisfying answer: the mobility tradeoff is true on a large scale and over the long haul. Along with time and distance scales and archery ranges, this assumption built into D&D seems to be imported wholesale from the larger-scale wargames both Gygax and Arneson were most familiar with.
So while a heavily armored fighter can indeed run around and do jumping jacks, they tire a lot quicker from that activity. And being able to sustain a pace is what matters for a unit-based wargame where turns are a matter of minutes.
So in a gaming context there are three situations where movement matters.
1. Exploration and long-distance travel. Over ten-minute turns, hours or days, fatigue and needing to rest would definitely slow an armor-wearing person to about half the move a non-armor-wearer.
2. Tactical movement in combat. Here,movement from one foe to another, to flank, and so on tends to be short and sporadic. I've noticed that movement rates in dungeon combat, even if cut short to reflect being cautious and the possibility of making an attack. In a 30'x30' room, a plate-armored fighter's six 5' squares are enough to cover just about any kind of tacical movement needed, and an unarmored 12 squares are just excess. So even though the lobster-plated guy is entitled to more because fatigue's less likely to come in, it probably won't interfere -least of all if you are using area positioning or "theater of the mind" to run combat.
3. Hauling ass. In chase situations, armor and load will determine who catches up or gets away, and while it makes a slight difference in timing whether this is due to fatigue or movement, the ultimate effect is the sme,
4. Charging. Again, realistically an armored fighter making a long charge might suffer a round or so less of arrows and spells from the defenders before closing than their low movement rate would indicate. But it's likely they would get there in less than full fighting trim. So, the slower movement here can reflect the fighter conserving energy.
In short, "realism" is often invoked as a reason to "fix" D&D but in this case I think the stark simplicity of the speed/armor tradeoff. If you want to cover short-term speed bursts I recommend ruling that you can move as unarmored in armor, but take 1 hp nonlethal fatigue damage per level each round you do so, that can be regained at 1 hp/level with each round of rest.