One choice is, of course, to simply chop or slice the leek as you like, put the leek pieces in a large bowl of water, swish them around to loosen and grit and dirt, and lift the pieces of leek out of the water.
The problem with that method is two-fold: first, you need to fish the leek pieces out of water (and no, you can't just dump them all into a sieve - the dirty water poured all over them in the process will redeposit the dirt!); second, if the leeks are quite dirty, the cutting surface will need a good scrubbing, too.Easier, faster, and cleaner is this method. It starts by cutting off and discarding the root end of the leek. Trim off as little of the leek white as possible; it's the part called for in most recipes.