Greek! The Romans themselves were Grekophiles, and most well-educated Romans would be expected to know Greek. Greek was the lingua franca of the eastern Mediterranean (and was also widely spoken in many western mediteranean settlements like Massalia and Emporion) and continued to be the primary language of communication there up to the fall of the Byzantine Empire. You can read more here.

Imperial Aramaic would be an example of a lingua franca pre-dating the use of these two languages in that capacity.

After and amongst the disruptions of the end of the Bronze age Aramean semi-nomadic tribes migrated throughout the Near East establishing new kingdoms and often overrunning existing ones. Babylon was sacked at one point and the Hebrew Bible is replete with references to Aram-Syria menacing Israel and Judah. They never were able to conquer Assyria (northern Iraq) though and ultimately Assyria brought most of the Near East under its control. As also referenced in the Hebrew Bible the Assyrians were quite fond of forced deportations and ultimately a lot of Aramean speaking peoples were settled in areas like Mesopotamia that already had a strong Arameans influence in certain areas. Ultimately, the dialect of Aramean spoken in Mesopotamia (Imperial Aramaic) became the "lingua franca" of the (Neo) Assyrian Empire. Assyria fell and was briefly replaced by a resurgent Babylon, which didn't knock Imperial Aramaic off of its pedastal since the Babylonians were speaking Aramaic by that point anyway. Once Cyrus the Persian overthrew the Babylonians and the Medes, he and his successors were practical enough to continue using this dialect in the administration of their empire. Koine Greek would steal some of the thunder of Aramaic after the conquests of Alexander the Great, but it was still widely used in the Near East at the beginning of the Common Era (the Christian Bible, while written in Koine reflects that Jesus spoke a form of Aramaic) and only stops being used as a lingua franca in parts of the Near East after the "Arabization" associated with the Muslim Conquests.

Yes, actually. One such language was Koine Greek.

One consequence of Alexander's conquests of Asia and Egypt was a process called Hellenization, the transference of Greek culture into the newly conquered territories. Greek architecture, philosophy, technology, literature, and religion were spread throughout the kingdoms that emerged in the wake of Alexander.

Koine Greek became a lingua franca, though obviously that term would not have been used. In the Middle East, Koine Greek was even still used as a common language after the arrival of the Roman Empire. This is actually the reason that the New Testament was written in Koine Greek: anybody who could actually read could probably read Greek, and those who couldn't read Greek would still have understood it when it was read aloud. It was this common language that helped spread Christianity throughout the Empire.

After the fall of the Roman Empire, Koine Greek continued to be used in the Byzantine empire as a written language (although at this point, the language was in a period of transition between Koine and Modern Greek, a period called Medieval Greek). It would only be a few centuries after that before Arabic was introduced to the region and Koine's dominance came to an end.

Nikolaos P. Andriotis, History of the Greek Language.
Elizabeth Pollard, Worlds Together Worlds Apart.