The most common and most likely answer to this crossword clue is the 5 letter word IONIA
I've seen this crossword clue in The Daily Telegraph
I
O
N
I
A
More clues leading to the result IONIA
- Brigadoon girl heading off round island's ancient region
- Ancient area in region I annexed
- Vacation I arrange includes ancient region of Eastern Mediterranean
- Ancient Greek region backing top class repeatedly
- One-on-one affair's beginning in a region of Greece
- Old Greek land in region I approached
- Region Greeks colonized in nation I admire
- Part of Asia Minor seen during vacation I arranged
- One inhabiting island, part of Asia Minor
Buddy explains!

The straight part
We know that "ancient region of Eastern Mediterranean" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case ancient region of eastern mediterranean = ionia.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.