I can not seem to gt rid of the really annoying BLACK that always seems to show up in Blender Cycles when rendering glass. I've even used a transparent shader to NO avail.
Here's a material fix that I applied that seemed to head toward what you were wanting. Same basic components, just re-arranged a little.
I think you can get your reflections back using a method something like this afterward:
If you want to get even more of the reflection back you can just keep the progression of your mix nodes, something like this: