We've now launched turn notifications for Frozen Synapse on Steam.
The next time you log in you'll be asked if you want to recieve notifications. If you choose yes you'll be notified by Steam itself as soon as a new turn is ready for your matches, even if the game itself isn't open. You can choose to turn these notifications on or off from the Game Notifications Settings
You can also launch the game from your notifications page and once you log in the relevant match will load automatically.