DO NOT TAKE A FIXIE. Do you have any idea of the type of hills you're going to encounter? I don't know too much about Europe, but I understand it's got some mountains. And once you get into the middle east, well, the entire Holy Land is nothing BUT godawful hills. You won't make it on a fixie unless you're Jesus. Go with a road bike with beefy tires (for those bad quality back roads in poorer countries) and low gears. Oh and don't go through Syria now, obviously.

Or do. People have probably biked through more dangerous places.