I noticed I could still move after carrying too much loot. Do I get some sort of penalty for carrying too much stuff?

Yes. It is dependent on your strength. If you look towards the top left of your inventory screen you will see how many stones you are carrying and what your encumbrance is, e.g. light, moderate, and etc. Here is a tip. Always make a follower you plan to keep carry the heavy stuff. ;)

I almost forgot to add. It makes you slower. Light adjusts the speed minimally, moderate even more so. When you have too much you act like a stiff post waiting to be shot for target practice.

But what are the penalties? If they're negligible I'll just ignore them.

Sorry, I edited my post while you replied back. You'll see it there.

I believe there are also accuracy penalties or something if you are heavily encumbered. But I might be mixing that up with just using a weapon under the strength requirement.

Yes sir you are. That would be an interesting addition however. It just makes your character slower thus eats up action points and makes kiting damn near impossible.

Copied from the manual

"When a Character picks up an item of sufficient weight to cross over to a new
level of Encumbrance, the player will receive a message in the Message Window.
There are several levels of Encumbrance, from light to heavy, and each level
causes the Character to move more slowly by progressively reducing his or her
Speed. At the highest levels of Encumbrance, a Character may take Fatigue
damage as well from carrying such a great weight. Under no circumstances can the
Character carry more in stone than his Carry Weight."

Ah! Ok! great, thanks for clearing that up!

Just to be clear... the lowered speed effectively means fewer action points in combat. Or, if you are playing in real-time, you will simply move more slowly, making fewer attacks in the same amount of time. I found that characters who want to wear heavy plate armor pretty much have to accept that they'll be lightly encumbered most of the time. It's up to you how highly you value your action points.