All of the "Legendary Dogs" flee so it's important you know where your going before you try to catch one.
 Original Appearances
The "Legendary Dogs" were first introduced in Generation II and you went on a hunt with a person named Eusine who is tracking down Suicune. You eventually catch up to Suicune and battle it after chasing it into so many routes that it just gives up and fights you at the end. The other Legendary Dogs roam freely through out Johto until you catch them. It's best to bring fast balls because Entei and Raikou are pretty fast when it comes to speed. In order to unleash the Legendary Dogs you must go to the Burnt Tower in Ecruteak City and go to the lowest floor. They should be waiting for you there (You must beat your Rival first), but you can't catch them there because they bolt loose after seeing you.
 Reprised Appearances
The Crystal Version, along with the Generation IV remakes Heart Gold and Soul Silver, had an interesting wrinkle to the formula for capturing Suicune. While Raikou and Entei would still run on sight, Suicune would be available to be captured at a later point in the game. In Crystal Suicune would appear at the Tin Tower after satisfying certain conditions, but Suicune would appear in Kanto in Heart Gold and Soul Silver before finally battling at Route 25.
In each of those instances Suicune would not run or use Roar.
 Main Games
One of the three Legendary Dogs would appear in the Generation III remakes Fire Red and Leaf Green. Which of the three that appears after obtaining the National Pokédex depends on the Starter Pokémon that you chose at the start of the game. If you chose Bulbasaur as your starter, Entei will start roaming Kanto; if you chose Charmander, Suicune will starting running wild; if you chose Squirtle then Raikou will start moving around the region.
Regardless the Legendary Dog that appears, they can still appear on the water while using Surf in the Generation 3 and 4 games. They will not appear in the Seveii Islands, however.
Raikou, Entei, and Suicune appear as a Shadow Pokémon with one of the Cipher Admins. Raikou appears with Cipher Admin Ein at the Cipher Lab, Entei is a part of Cipher Admin Dakim's team at Mt Battle, and Suicune appears when facing off against Cipher Admin Venus at The Under, a mining town near Pyrite Town.