The parts of the Empire that were thoroughly Romanized were mainly troublesome provinces that required a lot of ass-kicking—Britannia, Gaul, Hispania, etc. After watching a vast proportion of their countrymen from all strata of society slain or enslaved, the local elites that survived hopped on the SPQR bandwagon quicker than people who play Paradox games. The Roman assimilation squad also used what we'd consider "heavy-handed PR strategies" to further integrate the conquered: merging local deities with their Roman equivalents and showing off the superiority of Roman culture with fancy innovations like roads, bath houses, Latin, crucifixion and so forth.

The other side of the Mare Nostrum was a different story altogether. These were heavily urbanized areas that had belonged to successfully-administrated states for centuries (the Persian Empire, the Greek city-states, Alexander's empire, the successor kingdoms of the Diadochi, etc). They already had a culture, and it was considered equal (or even superior) to that of the Romans. The Romans didn't have to come in and hand out their brand of civilization to the conquered while building up the area; it was already built, densely populated, well-organized and efficiently taxed. The Roman governor arrived and a laurel appeared above the head on the local coinage. These provinces required no remodeling. They were move-in ready!

The common language of the people who wrote down the New Testament (and of the regions in which they lived) was Koine Greek. The Gospels, Acts, Epistles and Revelation are believed to have been written in the first century (generally between 50 and 110 CE, depending on which part), by both Jewish Christians and Hellenized Gentile Christians, mostly in Syria and Palestine. Roman dominion in these areas was relatively new, by the standards of the time, and one could argue that the prevailing Hellenistic culture of the East was the influencer, as it evolved into Byzantine culture and would remain Greek-oriented up until the Muslim conquests.