Elder Scrolls: Morrowind didn't have quite so immersion-breaking chests in the way you are thinking, but they still had ways of giving the player items in dungeons in ways that were consistent with that world.
Basically, dungeons fell into several categories, but the broader ones were:
Ancient ruins
- Dwemer (dwarf) ruins - mostly filled with junk that wasn't worth stealing (gears and such), but some ancient books that have side-quest significance, old armor which mostly sucked but could be sold for medium-ish prices, and gemstones, ancient pottery. The reason why they weren't entirely looted is because either their traps were still active and keeping out intruders, or because the ruins were occupied by some bandit group or vampire clan or necromancers-in-hiding.
- Daedric shrines - often occupied by cultists, who like placing gemstones on the altars. Sometimes the older Daedric shrines may have really valuable daedric armor. One of them is sunk under the ocean, and you need to use breathing potions or spells to reach.
More modern ruins
- Tombs (occupied by ghosts, skeletons, corpses), people buried their dead with gold and equipment and so on.
- Ruined castles (occupied by undead, bandits, vampires, or rebel soldiers), unless you find some secret room that the occupants missed, the only loot there would be the occupants' property.
Non-ruined structures
- Castles / Fortresses / Cities / etc... Don't let the guards catch ya stealing. But if you do get caught, you better be able to fight your way out. And know how to stay out of reach of the law from thence onward.
- Houses / Manors / Stores - Again, don't get caught stealing. But you can, if you're a good thief, or don't mind killing the occupant in such a way that he's unable to call for help.
Caves
- Mines - often occupied with regular NPCs who don't bother you unless you get caught stealing from them or attack them. You can steal from the miners (not that they had much of value), and you can also mine resources to sell from the mines. There were two types of mines in Morrowind; one gave ore (called "ebony"), the other gave a crystal material (called "glass") used by merchants to craft light-weight but durable armor and weapons. You can sell the materials.
- Nests - occupied by a specific type of creature, similar to termites I guess, that'd lay eggs that were eaten (?) by the people in the region. You can steal the eggs and sell those.
- Caverns - Occupied by monsters or (surprisingly often) by bandits or smugglers. You can loot the bandits' / smugglers' property after murdering them.
Basically, when looting stuff, the items would often be on tables, on bookshelves, on altars, and etc... not in containers, and when they were in containers, the containers looked more like foot-lockers (i.e. flat-topped chests that weren't anything fancy), or just crates and barrels, or cabinets/drawers.
And most anything good was being equipped by the NPCs, and used against you, until you killed them and took it, or pick-pocketed them. The bodies of your enemies were the chests.
Further, Morrowind had so much 'junk' items populating the world (bread, candlesticks, clothing, meat, apples, books, plates, forks, handkerchiefs, etc...) that it gave the world a lived-in feeling, so you weren't surprised at all that there were good stuff mixed in with the junk. And while two of the bandits just charged at you with rusty swords, it still made sense that there was a few minor magic scrolls nearby, because one of the bandits launched a fireball before moving in to engage, as if he was a minor magic-user still learning to use his abilities. And ofcourse if health potions and stamina potions exist in the world, bandits would certainly carry a few with them incase things went bad. As for those bottles of brandy in the city guard tower, those are purely for medicinal purposes, right captain?