“The origins of Indo-European languages is generally believed to be central Asia...”
I’m no expert, but in my reading there hasn’t been any general belief in that, though I’ve seen what you’ve mentioned also asserted - many locate it to the shores of the Black Sea in Europe, or the shores of the Black Sea in Anatolia.
I am certainly not an expert, but the lack of a common word for Ocean or See would seem to mitigate against a Black Sea origin. In Latin “mare” means sea or ocean, in Germanic languages, the word means marsh or waste land. http://www.etymonline.com/index.php?term=moor