I'd suggest Bernard Lewis's
The Middle East: A Brief History of the Last 2,000 Years, available for about ten books in paperback. It's a very well written book and it helped me in understanding the Palestinian/Israeli conflict.
There is perhaps one caveat which should be mentioned: Lewis is not popular with Muslims because he pretty much blames them for their own problems.
EDIT: In reading back over this I see it was the region's ancient history you were asking about. For that, I'd recommend Isaac Asimov's
Guide to the Bible. Because the bible covers so much of the ancient history, Asimov's book is, by its nature, a factual, secular history covering the same period of time. It is much larger than the Lewis work but it is also much more accessible.
[This message has been edited by berberry, 03-15-2004]