I had the same problem with my son. I tried the Nuby cups b/c the nipple is more like a bottle, but it is made like a cup with handles on both sides. We did that for a couple weeks and then moved to the Playtex brand of sippy cup that has a "firmer" nipple, that is rubbery, and still has the handles on both sides. He did really well with that. Also, try just offering him a cup with a straw in it. I wasn't sure my son was ready for that, and at first he didn't get it, but after a while, he thought he was "big stuff" to drink from the straw. That seemed to get him more used to and confident in drinking milk from a different source than the breast or bottle.
Also, if you are breast feeding, I have heard that if you put breast milk in the cup first then they make the transition easier???? We made the transition okay without doing that, but you might try it if you are??? Hope that helps!