I also had this problem and finally asked a therapist about it last year, because I was pregnant and having to do a million blood draws. He surprised me when he said the last thing you want to do is try to relax (which is of course what the nurse always tells you to do).

Instead, start tensing muscles in your body - legs, hands, wherever. This keeps the blood pumping to your brain so you don't faint.

It absolutely works. I can even watch them draw the blood now and feel fine.