Everywhere except in the US airlines charge fuel surcharges that are broken out separately. When you book an award ticket you have to pay with your miles, then pay the taxes and then also pay for the fuel. Just search around on the European fora. Other airlines than DL may only give you as little as 25% of miles flown when flying on economy tickets and may have higher cash required to redeem than DL, so the grass is not always greener elsewhere.

One advantage of programs such as Flying Blue is that you can book one way award tickets for half the number of miles which during promotion times means you only need 12500 miles to fly from Europe to the US to position yourself. By planning and timing your subsequent flights you can then just book only US departures with Skymiles to avoid the charge.