Catching Suicune and the other two is the hardest quest in the game. You need a few things:
~ a Pok�mon with Fly
~a Pok�mon with Surf
~ a Gengar with Mean Look and Hypnosis at L39
~ a Quick Claw (get it in the bug catching zone from the girl)
~ money money money and money.
First, give the Gengar the Quick Claw and buy 50 Ultraballs and 30 Max Repel. If you have a few Heavy balls, things get much easier. You need back apricorns from them, you get them south of Ecruteak city. Many people suggest Speed balls, but they are really crappy.
REMEMBER: Gengar has to be at L39, not 40. That's important!
1) Go to Route 32, 42 or 38. Yur best bet is Route 32, I view all three dogs there for the first time. Now put Gengar on top of your list and use a Max Repel. Walk into the grass now. You won't fight any other Pok�mon than one of the dogs. If you haven't seen one after three Max Repels, fly there again or go to another Route. Of course, they can be on every Route. For more info, go to "understanding her"
2) You've seen her? Now, first use Mean Look. If you're lucky, you're quick enough. If you're not, say bye-bye to Suicune, she'll run away. But if you use Mean Look in time, you have her - until she uses Roar. So put her to sleep fast and use Night Shade. Night Shade can't to double damage, so you won't kill her. If she is weak enough, use your balls. Your best bet is to use three or four Ultra balls first before using your Heavy balls. If you're lucky, you catch her.^^ If she wakes up and uses Roar, the battle is over, but she won't recover. Now, you can see her in the Pok�dex and things get easier now.
3) Look in your Pok�dex. Fly to a city, the best bet is always Ecruteak. It's said that she moves accidental, but believe me, it's NOT TRUE! It's a little tricky to understand her ways, so go to "understanding her".
When you finally make it and you find her again (don't forget: use the Repels!), do the same as above. I'm sure you'll catch her soon!
A hint: If you want another Suicune, take the first one with you in your new game. When you free the new one now, you can look for her in the Pok�dex now, you don't have to find her first!
CRYSTAL:
In Crystal, you won't have to search for her. She'll appear in the grass and this time, you SEE her. After you have seen her three times finally, she'll went to the Tinn Tower in Ecruteak City and you have to fight her there. The best: She's on the first floor, you don't have to climb the tower first. Houou isn't there, but Suicune is ways better, right?