What's the most efficient Minecraft mining strategy?

Question

When I'm not exploring fresh caves or lands in minecraft, I'm busy trying to find deposits of diamond, gold and iron deep in the earth. So far I've just been digging around just above bedrock on a whim to look for these rare minerals but I've started thinking that I should be taking a more controlled, efficient approach to mining.

What is the optimal mining strategy that yields the most rare minerals while minimizing blocks removed and time spent digging and traveling?

Edit: The existing answers are great for mining, but no one has addressed travel time to and from the mine which is usually ~60 blocks downward and incredibly tedious. Solutions?

Answer

Mining

Branch mining is the most efficient. Someone did a mathematical analysis on it in the official forums.

Here's a brief description of branch mining:

You dig down to the bedrock, because that's where you're going to find the most different kinds of stuff. One or two layers above the bedrock, so you don't have it interfering with your work, you dig a long tunnel, 2 high and 1 wide, which will serve as the trunk. Then, every 4 squares on either side of the trunk, you dig perpendicular tunnels extending out 20 blocks (or as far as you want). You leave 3 undug blocks between because veins are generally 2x2 horizontally, so you will see ore in the vein even though there is technically one row of unexplored blocks. When you finish that layer, you can start one a layer above, offsetting the branches by one block so that you are more likely to find veins you could possibly have missed the first time.

So essentially, a long hallway (the trunk) with perpendicular hallways (the branches) coming off either side.

█▓▓▓ █ blocks you should mine
█▓▓▓ ▓ blocks you can see
▓▓█▓ ▒ blocks you can't see if you don't mine ▓ blocks (there's none!)
▓▓█▓ Repeat this pattern as many times as needed

From a head on view

Travel

As with any mine, travel is most easily accomplished using minecarts. You can place a booster to get your cart traveling at max speed in a short distance so that you can make the trip up quickly, and you can even attach chests to mine carts to carry all your findings from the mine up the track very quickly.

Short of minecarts, make sure you're using stairs and ladders to their fullest potential. You don't want to be jumping up one block at a time all the way from your mine to your base.

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