Skipping them will reduce your risk of breast cancer by reducing your exposure to radiation. Here's the explanation for how that happens, from the Cancer Prevention Coalition:
Thus, premenopausal women undergoing annual screening over a ten-year period are exposed to a total of about 10 rads for each breast. As emphasized some three decades ago, the premenopausal breast is highly sensitive to radiation, each rad of exposure increasing breast cancer risk by 1 percent, resulting in a cumulative 10 percent increased risk over ten years of premenopausal screening, usually from ages 40 to 50 (4); risks are even greater for "baseline" screening at younger ages, for which there is no evidence of any future relevance.
If that's not enough to give you pause, look a little further at the Coalition's web site and you'll see that compressing a breast in which cancer is already growing can actually cause cancerous cells to spread.
There are all sorts of other issues with mammography, like poor quality control and the mystery of why the risk factors for breast cancer apply to white women but don't work well to predict the disease in black women.
These are the kinds of questions we should be making noise about instead of rejecting sound scientific advice about when--and whether--to have mammograms.