When I saw 2 escorts from AM earlier this year I used my real name. When speaking to the operator on the phone I used my real first name and surname because I wanted the girl to come to my hotel room and they like to have the name you are booked in with.
Besides, the girl having your real name makes things seem more friendly and personal anyway. If you want to chat to them, they may use your name when talking to you. If you've given them a fake name it would feel strange to hear it. If you want a more personal experience it can't really be obtained by having them call you by a fake name.
I can't honestly think of any problems that you could have with them knowing your real name anyway. It's probably impossible for them to try and blackmail you in any way if you don't live in Japan. They can't really do anything with your name against you. Maybe if you got into a lot of trouble, like say doing something to really upset the girl, it might cause you problems. But then it's not like you are going to go out of your way to cause trouble.
If you are really nervous about using your full name, have the escort go with you to a love hotel and then you can either just use your first name only.
I'll go ahead and endorse what JustSomeGuy said above.
There are two situations:
1) Don't use your real name = Meet at love hotel. Usually, just a Japanese cell phone would be required for contact purposes.
2) Use your real name = Normal hotel, for contact purposes. Hotels generally require your full check in name in order for calls to be forwarded to your room.
There is one small exception to #2: You usually can register an alias for privacy reasons. Most of the big name hotels will do this for their clients when requested. This is something that celebrities or other public figures do to allow calls to their room.
What you should not do, when avoidable, is give out credit card or billing address information connected to a credit card you used. There are some agencies that accept credit cards, but I would personally advise against using a card. Cash is much better in all cases.