the region that was to become known as Assyria

In prehistoric times, the region that was to become known as Assyria (and Subartu) was home to a Neanderthal culture such as has been found at the Shanidar Cave. The earliest Neolithic sites in Assyria were the Jarmo culture c. 7100 BC and Tell Hassuna, the center of the Hassuna culture, c. 6000 BC.

The Akkadian-speaking people

The Akkadian-speaking people (the earliest historically-attested Semitic-speaking people) who would eventually found Assyria appear to have entered Mesopotamia at some point during the latter 4th millennium BC (c. 3500–3000 BC), eventually intermingling with the earlier Sumerian-speaking population, with Akkadian names appearing in written record from as early as the 29th century BC.

During the 3rd millennium BC

During the 3rd millennium BC, a very intimate cultural symbiosis developed between the Sumerians and the Akkadians throughout Mesopotamia, which included widespread bilingualism. The influence of Sumerian (a language isolate) on Akkadian and vice versa is evident in all areas, from lexical borrowing on a massive scale to syntactic, morphological, and phonological convergence. This has prompted scholars to refer to Sumerian and Akkadian in the third millennium BC as a sprachbund. Akkadian gradually replaced Sumerian as the spoken language of Mesopotamia somewhere after the turn of the 3rd and the 2nd millennium BC (the exact dating being a matter of debate), although Sumerian continued to be used as a sacred, ceremonial, literary and scientific language in Mesopotamia until the 1st century AD, as did use of the Akkadian cuneiform.

Letter sent by the high-priest Lu'enna to the king of Lagash

Letter sent by the high-priest Lu'enna to the king of Lagash (maybe Urukagina), informing him of his son's death in combat, c. 2400 BC, found in Girsu.
The cities of Assur and Nineveh, together with a number of other towns and cities, existed since at least before the middle of the 3rd millennium BC (c. 2600 BC), although they appear to have been Sumerian-ruled administrative centers at this time, rather than independent states.