Anyone who reads about the last of the Russian Tsars and the Russian revolution is familiar with a disclaimer about dates. Usually the author uses dates New Style, but occasional dates are given Old Style. In 1918, after the revolution, Russia changed from the Julian to the Gregorian calendar, as had most of the rest of the western world much earlier. (George Washington, for instance, was born on February 11 but with the British adoption of New Style when he was 21 years old his birthday became February 22.) Eastern Orthodox churches still use Old Style for their ecclesiastical calendar. Thus historians have to ask which calendar is being used.
Luke was Greek. When he wrote his Gospel he clearly began it by using Roman dating. Throughout the book of Acts, though, he uses Hebrew date references even though his primary audience is clearly Greek.
The most obvious early reference is to Pentecost, and that is understandable. “And when the day of Pentecost was fully come, they were all with one accord in one place.” (Acts 2:1) Mention of the Jewish holiday sets the time and place, and explains why there was such a large crowd in Jerusalem. There was really no other way to date the beginning of the church.
Now about that time Herod the king stretched forth his hands to vex certain of the church. And he killed James the brother of John with the sword. And because he saw it pleased the Jews, he proceeded further to take Peter also. (Then were the days of unleavened bread.) And when he had apprehended him, he put him in prison, and delivered him to four quaternions of soldiers to keep him; intending after Passover [Easter in King James Version] to bring him forth to the people. (Acts 12:1-4)
It might even be understandable that Luke refers to the Passover here. After all, that was Herod’s specific timeline. Luke didn’t choose to mention that date; Herod did. Sometimes circumstances demand the use of a different calendar. After all, who would know the date of a Chinese New Year celebration just using the common Gregorian calendar?
Yet again, Luke chooses to mention a date based entirely on the wishes of the person about whom he writes. “For Paul had determined to sail by Ephesus, because he would not spend the time in Asia: for he hasted, if it were possible for him, to be at Jerusalem the day of Pentecost.” (Acts 20:16) When someone else specifies a date in a different calendar, you might as well repeat that date.
The one time that Luke mentions a date in the Jewish calendar independent of specific circumstances or people is worth noting. “When sailing was now dangerous, because the fast was now already past, Paul admonished them.” (Acts 27:9) It might have made sense to a Greek or Latin audience to use the ides of October rather than “the fast.” Nevertheless, that is a very specific and accurate way to describe the situation. The fast (Yom Kippur) occurs in early October by our calendar. It marks a clear beginning to the Medicane season. Winter storms on the Mediterranean can be very severe, with winds reaching near hurricane force. Because it is a shallow sea, the effect of the winds is amplified by the waves. Neither Paul nor Luke were known to be sailors, but they knew that once the Jewish fast was over it was no time to be setting out on the open waters. The sailors from Alexandria apparently did not know this or thought they could handle it. A fourteen-day storm is unusual, and may have been God-influenced.
Another way of looking at all this, though, may be reconsidering Luke’s audience. He addresses the book of Acts to Theophilus, a Greek. Whether that was the name of a real person or not is open to interpretation; still, it is Greek. Nevertheless, one of the principal messages of the book of Acts is that the gospel is available to both Jew and gentile. In that, it could be argued that Luke’s principal audience was the Jewish people, whom he was trying to convince. It makes sense, then, that he would use Jewish holidays as date references.