Ginger, the Academy of Sciences does not agree with you on this one. Low level radiation is now pretty much agreed by scientific consensus to work just the way you say it does not being hit by one hammer over 30 months is about the same as being hit 30 hammers in one month. There maybe a little difference - maybe being hit by one hammer once a month for 30 months is like being hit by 20 hammers. But not a lot of difference from linear [link]
Calculations in this report suggest that approximately one cancer (star) per 100 people could result from a single exposure to 0.1 Sv of low-LET radiation above background average, assuming a sex and age distribution similar to thatof the entire U.S. population, the BEIR VII lifetime risk model predicts that approximately 1 person in 100 would be expected to develop cancer (solid cancer or leukemia) from a dose of 0.1 Sv above background, while approximately 42 of the 100 individuals would be expected to develop solid cancer or leukemia from other causes. Lower doses would produce proportionally lower risks.
So standard assumption as of 2006 was the Linear Threshold Model, the one you are rejecting. Since Academy links are trick, if it does not work you can google " Health Risks from Exposure to Low Levels of Ionizing Radiation:BEIR VII Phase 2 (2006)" Now there are dissenter - quite a number of them arguing that there is indeed a threshold that below it the body is self healing and radiation may even have a homeopathic effect. Similarly there are dissenters who say that below a certain level the bodies defenses fail and that effects are even greater per fraction of a Sievert at very low doses. The linear No Threshold appears to be the conservative estimate at the moment. I know the French Academy of Sciences disagree, but at any rate that the LNT is false if far from settled science.
More recent EPA 2009 [link] That is the "One hammer per month for 30 months is like 20 hammers" version - That is at low doses the effect is 1.5 less than at high doses. Still pretty close to linear.