The Assyrians are a good civilization for early rushes since their villager bonus allows them to scout the area faster than other civilizations.
The Assyrians can begin to flex their muscles as soon as they hit the Tool Age and build an Archery Range.
Their fast villagers allow to get ahead of other civilizations, both in research and resource gathering, so you can try to be the first to make it to a subsequent age, build a small army, and strike while you have a technology advantage.
The Assyrians also have excellent archers and will make up a large part of your army. It's not a bad idea to get a small group of archers together immediately after you get to the Tool Age and build an archery range.
Use them to harass and slow your enemy's resource gathering, picking off villagers and retreating when they come under fire.
With their fire rate increased, a large amount of them can wipe out an entire village within minutes. This makes the Assyrians an excellent choice in small land maps.
Don't depend entirely on your archers - you'll still need a varied army.
In major battles, use infantry to engage enemy troops, so your archers can stay behind and fire their arrows.
The speed advantage of your villagers will be blunted in a long game. Other civilizations will be able to field a more technologically advanced Iron Age force given time.
Don't hedge your bets on controlling the seas either.