Was The Arabian Desert Once Lush and Green? (Like the Sahara)

I understand that thousands of years ago, much of the present-day Sahara Desert was wet and green. It had rivers and lakes, and trees. The Sahara dried up and became a desert (because the Mediterranean Sea cooled and ceased to provide moisture for rain over North Africa).
Was the Arabian Peninsula always a desert? Or was it once nice and wet and green also?