The language was centered in Mesopotamia (what is today Iraq). This is essentially the fertile alluvium between the rivers Euphrates and Tigris south of Djebel Hamrin, comprising approximately 15,000 square kilometers. The capitals of the area are the cities of Akkad = Babylon in the center (near present-day Baghdad) and Assur in the north of Mesopotamia. When the Assyrian empire expanded, Niniveh instead of Assur became its capital.
Map from http://www.fsmitha.com/h1/map03sar.htm
At the time of its greatest extension, around -660, the Assyrian empire was much larger, and Assyrian was widely used in the Near and Middle East.
Map from http://karenswhimsy.com/assyrian-empire.shtm
A time-line with historical maps is on the website of ECAI Iraq.