Well according to [1] crew of the Concord would get 50-150mSv/yr. US regulation is that a radiation worker can get 50mSv/yr and EU is 20mSv/yr. So this is clearly not acceptable.
Your intuition about carbon-fiber is correct.
What I would guess they are doing though is a layered system, as it is hard to make up the difference with just dopants. Other materials are good too, like Kevlar [2]. So you layer these materials together and get better overall protection. Lots of research being done for this right now actually. NASA has some SBIR phase II funding for low weight, structural radiation shields. Though on Earth you can get away with a lot more than you could in space (launch weight).
[1] http://www.aviationdoc.com/articles/Radiation%20and%20Aircre...
[2] http://esmat.esa.int/materials_news/isme09/pdf/3-ground/s3%2...