The easiest way to catch - in your case - Raikou, is a very close method to what muffineater said. To simplify it, however:
+ Wobbuffet is your friend. It's ability Shadow Tag prevents Wild Pokémon from fleeing battle (as well as switching when in trainer battles.) That takes care of Raikou's habit of running away as soon as you see it.
+ Repel, Super Repel, and Max Repel are put to their best use here, as well. Since Raikou will appear at Level 50, it's a good idea to have your Wobbuffet close to, but under that level. This will make sure that you won't encounter any Pokémon under Wobbuffet's level, so that the chances of a Wild Pokémon being Raikou are increased (as far as the actual encounter rate goes.)
+ ANY PokéBall works. Obviously, your best bet is to use the MasterBall, since you'll not have to worry about anything other than actually finding the beast. Ultra Balls seem to be the most popular, with their increased Hold Rate.
+ Raikou (as well as Entei and Suicune) will ONLY be found on the Mainland. They will not appear on Knot Island, Chrono Island, Quest Island, or any of the Sevii Islands.
The best thing to do is stock up on PLENTY of Ultra Balls, and it wouldn't hurt to have some Great Balls and PokéBalls. I've found that when Ultra Balls start holding it to the third count (Breaking free right before it would have normally caught), throwing lower grade Balls (ie PokéBall and Great Ball) seem to have a bit higher chance.
With that in mind, your first goal is to encounter Raikou. The best way to do this is to go to Pallet Town, and surf to the south to that small patch of Grass. Put Wobbuffet in the first slot, spray some Repel, and start searching.
If you get lucky enough to encounter it, DO NOT LET IT USE ROAR. Apparently, there's a small glitch in (some of ) the games that, when you encounter one of the three Legendary Beasts, if it uses Roar in battle, it's gone from the game and you won't find it again. I can't speak on behalf of all games, however, but I do know of several instances that this has occured, my own game included.
My recommendation is, if at all possible, find someone who would be willing to let you borrow their Raikou. Have them trade it over, and then you can trade it back. Since you didn't catch the one in your game, you'll be able to see where it's located in Kanto.
Then return to the grassy patch, and if it's not there, surf up and back between routes until you find it. Then you're on your way.
If you can't, however, use that method, then the easiest way I've found is to simply not look for it. You'll find it quicker when you're looking for something else and not expecting to find it at all. Just try to always have a good stock of assorted 'Balls, Repels, and Wobbuffet in your party.
Looks like there's no need to keep this open, so I'll close it.