Some useful definitions:
http://www.world-nuclear.org/info/inf51.htm

On topic and also for those worried about depleted Uranium:
"The isotope U-235 is important because under certain conditions it can readily be split, yielding a lot of energy. It is therefore said to be 'fissile' and we use the expression 'nuclear fission'.
Meanwhile, like all radioactive isotopes, it decays. U-238 decays very slowly, its half-life being the same as the age of the earth. This means that it is barely radioactive, less so than many other isotopes in rocks and sand. Nevertheless it generates 0.1 watts/tonne and this is enough to warm the earth's mantle."
[as my brother noted in a consersation "the depleted uranium rounds would be less radioactive then the sands they are lying on."
http://www.world-nuclear.org/info/inf14.htm

As far as Barry's concerns in general terms I would say that the Moon (or Mars) material is not any more radioactive then Earth material- there could be a possiblity that in general the Moon is less radioactive then Earth:
From: http://hps.org/publicinformation/radfactsheets/radfact25.html
"Small amounts of radon, a radioactive gas which comes from the radioactive decay of uranium, seep into the atmosphere from the soil. On average, inhalation of the radon in homes and other buildings accounts for 200 millirem per year. This is about 55% of the total radiation dose an individual receives in a year from all sources, including medical x rays, cosmic rays, building materials, the earth's crust, and ingested radioactive materials."

Please don't mistakenly assume I mean that the total radiation dose from being on the Moon *would* lower then living on Earth, I'm saying the radiation dose due from a radioactive ground *could be* lower. [Not including the radiation from the Sun and galactic sources].