My name uses alternating “ah“s and “ee“s to make “AH-dree-YAHN Ee-VAH-kheev“, where the “ah“s are halfway between a long “aaaww” and a short “a”; the “dree” includes a rolled “R” — which any English speaker can say: just say “ditty” really quickly, minimizing the first syllable to almost nothing; and the “kh” is a softer version of the sound you make while gargling (as in the Scottish “loch”).
The result sounds like this:
When saying my first name in Ukrainian, some accent the first syllable (“AHD-ree-yan”), while others accent the third (“ad-ree-YAHN”). Either is fine. And for the record, my full given name is Adrian Jurij (or Iurii) Mykhailo Ivakhiv.
But if it’s easier for you, “Ay-dree-yen” (or “Eh-dree-yen” for Canadians), as in the movie “Rocky,” is fine, too.
Here’s the general rule, then: If you have any difficulty rolling your R’s, please use (Rocky’s) “Adrian” as it would normally sound in English. If you are a native or learned R-roller, then the original pronunciation (as described above) is best.