Just like Whispers of the Old Gods, you can get some free Mean Streets of Gadgetzan cards just by playing.
Hearthstone's latest expansion pack: Mean Streets of Gadgetzan launched today. But what if you don't have any gold or money to buy card packs? Fret not, as just like with Whispers of The Old Gods Blizzard is giving out a few packs for free. The first hit is always free!
Upon launching the game, you'll be offered the following quest: "Join the Goons - Win 3 games as a Hunter, Paladin, or Warrior." Completing it will reward you with two card packs, and the next quest in the chain: "Join the Jade Lotus - Win 3 games as a Druid, Rogue, or Shaman." This quest will also reward two packs when completed and unlock the final quest in the chain, which unlocks an additional two packs when completed: "Join the Kabal - Win 3 games as a Mage, Priest, or Warlock."
So all up, you can get six free packs. It's not a lot but hey it's something!