How can I prevent griefers from grabbing all my loot in Minecraft?

The Ender Chest is the best approach, but if you don't want/can't use an Ender chest there, here are some solutions I have prevented theft in Minecraft.

Redstone Traps

I have in the past created a storage room with tons of trapped chests, ~25% of which were "real" and the other ~75% killed everyone in the room. Fall damage is the best in my opinion, because it is hard to prevent once that trap is triggered, enables you to collect the thief's items, and your own items aren't destroyed if you trigger your own trap. Lava takes time, and the thief can break/get out, and his items are destroyed. Don't use TNT because that destroys your items. This usually only works once though, and trapped chests are suspicious. I like this approach because I am a Redstone kind of person, and I like the challenge of creating red stone traps for the thieves.

Hiding Your Chests

Obtain stone blocks. Go down into your mine. Mine 5 blocks down. Place chests and valuables. Write down coordinates. Place 5 stone blocks above. Tell no one that you even hid a chest. This method has always worked for me, and it is cheap.

Having the item in your inventory

Items in your inventory can't be stolen when you are offline. They can kill you, but you will know who did it, and can talk to them/take revenge depending on your situation. I like storing all my valuable diamond armor on me. Swords too.

As a side note, I usually never have all my diamonds in one spot. I distribute my diamonds among chests, and when a thief is successful, they usually leave after they steal one set of diamonds.

You can use an Ender Chest to store things that you don't want stolen. The contents of the chest are unique to the person opening it, so a potential griefer won't see anything that they haven't put in themselves.

The recipe for an Ender Chest is as follows:

Crafting recipe for an Ender Chest showing 8 blocks of Obsidian arranged in a square with an Eye of Ender in the middle

^{Image courtesy of MinecraftXL}

There are a few important things to note about Ender Chests:

They do not drop their contents when broken like regular chests do
Every Ender Chest shares the same inventory, so while this means you can access the contents from any Ender Chest anywhere in the game the amount of available space is extremely limited
Ender Chests require a Silk Touch pickaxe to pick up without destroying them
Unlike standard chests they cannot be joined together to increase their capacity
You cannot share their contents with another player, so if you need to do this an Ender Chest is not appropriate

If Ender Chests don't meet your requirements for whatever reason then you're going to have to look at alternative solutions such as physically hiding the chest that your valuables are stored in. The most important thing to remember about this approach however is this:

If you can access the chest then so can any potential thief!

This means that there is no guaranteed method of securing a regular chest against griefers, all you can do is make it unreasonably difficult for them to gain access to it.

If you're the server host you can also get mods such as Lockette or Light Weight Chest Protection that allow you to make chests private, but that's potentially a quite heavy-handed solution and might not be the right thing for you.

Hide your items cleverly

Disclamer: Xray is used for demonstrational purposes here. This kind of cheating is bannable! Do not try this at home! We are what people call professionals. That means that we tried it at least 3 times.

Hiding your items at a randomly placed chest underground will protect your items only partially; cheaters may use mods/texture packs/hacks (generally called "Xrays") to see through walls and find your chest(s). For example, this is how a randomly placed chest, as other answers state, seen by xrayers:

enter image description here

But if you hide your container at the proper place, it won't get revealed:

In Lava:
- it is opaque; it can't be seen through. Ideally, place your storage under a natural lava pool/cave. Also, nobody will run around checking around lava pools for diamonds.
Encased in cheap resources:
- encase your storage with iron/coal ore. Cheaters usually go for chests (loot), diamonds (loot) and people's livings (more loot), not for coal.
At a very low level: Place your storage as low into the bedrock layers as possible, then enclose it with lava/coal/iron/all at once

But wait! What about chest highliter mods/hacks, which show you where the closest chests are?

Well, this one is simple: Don't place your items in a chest. Use clever containers, like:

Droppers (up to 9 stacks of items)
Hoppers (up to 5 stacks). Plus, hoppers blend in with Bedrock and aren't usually visible with xrays
Furnaces (up to 2 stacks)
Donkeys or Minecarts (up to 27 stacks)

Using Hoppers is the safest option, for multiple reasons (same color as bedrock, not entities, usually filtered out by xrays)

Combine with other answers!

I combined the methods and tips from other answers (mainly my own). Here's how to keep your items safe:

Find a cave flooded with lava as it isn't transparent - blocks x-rays, as seen below.

But that still isn't good enough! People can turn off lava visibility!

Well, yeah. Then your storage is visible. The next step solves it, though.
Dig under it, and fill it's floor with some cheap resource, preferably Coal Ore - blocks x-rays if lava is "turned off":

Here's shown how I hid it (with part of the coal uncovered):

Wait, what if they have Lava and Coal ore turned off, as in the image below?
Find a remote place in between bedrock blocks under the lava.
Craft many Shulker boxes and put your items inside (optional step)
Hide your items, by one of the following methods:
- Craft a pair (or more) ender chests; Place one down and put your items inside. Optionally place one ender chest to your hiding place.
- Place a hopper in your remote area and place your items inside.(Recommended by own experience. Xrays track Ender Chests more often.)
To sum up the 3 steps described above, here is how your hopper/ender chest hiding should look like from up close and from afar:
Add a few blocks of Lava above it to prevent accidental finding, and if, to possibly kill the finder. On the following image, a hopper is placed just diagonally (down to the right) from the lava:
Why not use a chest anyways? They'll find the storage if they turn off Bedrock, Lava and Coal ore at once. Then it's pretty obvious, whatever the storage you have.

How to deal with this? Well, chances are, the potential robber will have hoppers also off, because it isn't a useful block to keep non-transparent: