I've tried two or three different methods, and a single one stands out for its output.You'll need nothing more than what you would typically take on a mining expedition, apart from the addition of a bucket of water. Firstly, mine (or explore) to the general level of lave generation within the bottommost stretch of the Overworld, just before hitting bedrock. Once you've located a large pool of lava, get above it and place the water from your bucket on an adjacent or overhead block. Recollect the water, and repeat. Once all of the lava has been exhausted and you're left with a floor of obsidian, use this as your canvas to mine the area.
As this level is fairly easily accessed and close to the bedrock level, diamonds are sure to be found. Oftentimes, the removal of lava exposes diamonds above the pool or to the side that normally wouldn't have been seen.