The D&D version is described in great detail in Ed Greenwood's classic Dragon article "The Ecology of the Mimic", and, as with the article as a whole, the mimic's locomotion is a brilliant balance of ridiculously silly and seriously plausible.
I don't have access to the magazine,1 but from memory:
Mimics are amorphous beings. They don't spend all day looking like a chest; they see or hear adventurers coming, and mimic a chest because they're smart enough to know that's how you catch adventurers.2 But they're not "some sort of slime-thing that's using a chest as a shell", they actually form their hides into rectangular shapes that look and feel as tough as wood, metal, or stone.3
They can rapidly form and extend pseudopods. Hence the famous Monster Manual drawing:
They can also secrete and dissolve a strong adhesive at will.
The way they move is by throwing out a pseudopod, covering it with glue, dragging their body over, and dissolving the glue as they toss out another pseudopod for the next "step". They can move about as fast as humans this way. This also means they can travel on walls and ceilings as easily as on floors.3
Also, most people forget that D&D mimics are intelligent, and will usually try to con the players through fast-talking instead of attacking them. And, while this is probably even less relevant here, they're immune to alcohol but have been known to feign drunkenness to trick adventurers.
So, how's that for an image: a mimic drunkenly dragging itself around the floor and the walls pseudo-fist by pseudo-fist, trying to make a deal with the PCs.
1. It really is worth reading, but I don't know where to find it. I mean, I'm sure you can find a pirated copy somewhere online pretty easily, but I don't want to encourage that. Go to the library, maybe?
2. They mimic other things in other environments. A statue in a city so they can eat people tossing coins in the wishing well, a bundle of war pennants in an orc camp, etc.
3. And how do they decorate themselves to have the right color and texture? I don't remember the details, but there were details. Something to do with pigment sacs and a special circulatory system, and also multiple rapidly growing layers of skin.
4. Yeah, that last part doesn't make sense—it should be harder to pull yourself up than sideways—but then AD&D never really got gravity.