Saudi Arabia is issuing tourist visas, but don't apply if you're Jewish or have traveled to Israel, notes Volokh. Don't forget some important restrictions for women, too.
Ditto for Iran. And, let us hope the US does not go in this nonsensical direction with its restrictions aimed at stopping terrorists. I had Ahmed Rashid's Jihad: The Rise of Militant Islam in Central Asia in my backpack while flying to DC the other week. And, boy, was I feeling queasy when they started going through it.