The problem is that Akkadian speakers in this period were largely sedentary, and many of the Aramaean tribes were nomadic, like the Arabs that succeded them. The Assyrians did attempt to resettle them, all over their empire, but this didn't have the effect of assimilating them - quite the opposite, actually. There were already significant Aramaic-speaking populations across the north (including Assyria, especially in the Khabur river triangle), and these efforts just had the effect of spreading them out. Also, there was obviously a natural migration of the Aramaean tribes to more fertile grounds, such as Babylonia. Remember that Abraham, who lived in Ur, was an Aramaean. Before long, you find Aramaic inscriptions in the major cities of Assyria, especially incantations, reflecting the extent to which Aramaic had become the popular tongue. Then the Achaemenids came and basically just recognized the facts on the ground.
By the reign of Hezekiah (end of the 8th c.), Aramaic was already the international language of diplomacy and commerce, and any member of the elite (among the Assyrians and their subject peoples) could be expected to speak it, as the account in 2 Kings demonstrates.
How do we prevent this? I have a few ideas:
No policy of deportation. If the Assyrians don't resettle Aramaeans everywhere, Aramaic won't be spoken everywhere, and therefore its potential to be a surrogate for Akkadian will be much diminished. That still leaves the burgeoning Aramaean populations in Assyria and Babylonia, though. I could envision a policy of excluding nomadic Aramaean tribes from the Mesopotamian floodplain, which would be within the abilities of a powerful centralized state;
Language reform. Standardize the script, make the written language reflect the spoken language to a greater extent, develop a form more suited to parchment than clay tablets. These things periodically occurred in Egypt but Mesopotamia was rarely unified long enough for such reforms. One of Aramaic's comparative advantages to Akkadian was that it could be mastered fairly quickly and you could encode far more data using smaller, less bulky media. Akkadian has prestige, however. The example of China is instructive; the Chinese script shares many of the same deficiencies as cuneiform (it was adapted to paper, obviously, which was a comparative advantage, but it was also considerably more complex than cuneiform, with a much larger repertoire of signs) but never succumbed to the languages that were written in alphabetic scripts that it encountered by dint of sheer numbers and prestige.
A coordinated policy of assimilation. Adapt the cuneiform script to other languages of the empire, so that they are all using the same repertoire of signs (much like Hittite and other languages were written using the same script). Again, the example of Chinese is instructive, as the Chinese script unifies all of the languages conventionally called "Chinese dialects" as well as many other unrelated languages. Encourage a standardized form of Aramaic written in cuneiform and teach it alongside Akkadian in scribal schools in areas where Aramaic is spoken. Hopefully the prestige of the script and its utility in unifying the region will retard the progress of the alphabetic script (because once you strip Akkadian of its script, it loses its prestige, and will probably become replaced by the languages of exploding populations like the Aramaeans). Cuneiform has religious prestige and social cache, as well as an immense literature, and Akkadian will need to build these things in order to survive. One possibilty, to kill two birds with one stone, is the establishment of an empire-wide examination system for the scribal schools along the lines of the Confucian examination system in the Middle Kingdom, which would help create a uniform civil service thoroughly indoctrinated in the Akkadian classics, thus ensuring that Akkadian will remain THE prestige language for as long as the empire stands, and giving people a practical reason to learn it. If this happens, it won't matter even if Aramaic overtakes Akkadian, because the new Aramaean overlords of the empire will have a vested interest in perpetuation the system.
The end result? A stable (yet multilingual) empire, unified by the cuneiform script, the Akkadian "classics", and a professional civil service.