I remember hearing it attributed to one of the 30s-50s literary humorist set, like George S. Kaufman or Robert Benchley, or maybe an Algonquin Round Table type. It helps to have had a good classical education, although Medes and Persians are mentioned in the New Testament. What wouldn't be obvious from the NT, though, is that the two were neighboring rival kingdoms.

The one thing it can't be is from Ancient Greek, or actually from any foreign language, because it's playing on the proverb "one man's meat is another man's poison" which is only a few centuries old, and the meat/Mede & poison/Persian homophones wouldn't work in other languages, as the words would be quite different.

But I can imagine Ogden Nash doing it, too!