Instructions
Walk around the world map and explore every section of the tall, wild grass that you can find. It can take a while, but eventually you will encounter and battle with the third Legendary Dog, whether it be Suicune, Entei or Raikou.
Use an attack like Hypnosis to put the Pokemon to sleep and stop it from running away. Battle the third legendary dog until its health bar is red. If you're worried about killing the Pokemon, you can use a low level Pokemon to attack and very slowly lower the legendary dog's health until it's in the red.
Throw a strong Pokeball, like the Ultraball, at the third Legendary Dog Pokemon to attempt to catch it. It will take some time, but eventually you will succeed and obtain the third and final legendary dog.