In general (and I'm not going to be specific because I know less than nothing about opening theory), you probably need to do a better job of creating threats and counterplay. A pawn storm takes a while, so if you are having this problem regularly it would indicate to me that you are not doing enough to keep white busy and/or uncomfortable.
An example game or two would help with the diagnosis.
It seems to happen to me every single game that I play as black, and it really annoys me. I was wondering if there is a black opening against e4 that does not have any risk against this sort of thing.
(Right now I play the Pirc and the Najdorf Sicilian)