First, Nanci is totally right. NO RADIATION with MRI. It uses extremely strong magnets to align the water molecules in your cells to get a "bounce back", for the lack of a better lay-term, that it turns into an image with some impressively intricate algorithms and programming. CT is like little, thin slices of x-rays. It's a finely controlled beam compared to general radiography.
Mammos use an extremely tiny amount of x-ray to get an image because it's only going through a relatively thin layer (few inches) of soft tissue. It also uses a highly sensitive image receptor compared to regular x-ray cassettes which allows for the much reduced dose.

What you may not think about is that you've lived in radiation since before you were born. The sun, earth and even our own bodies produce radiation. The amount you get from a mammo equates to about what you get if you hang out in the backyard (or on your bike ) for a few short hours. Your body is used to radiation. If it wasn't, no one would make it past childhood.
That said, yes, there is essentially no safe dose when it comes to radiation. Even small amounts could potentially damage a cell. But it so unlikely to happen from radiography that I'd rather "take a chance" and get a mammo or other test if it meant catching something early when it can be treated more effectively. It would be great if something better and safer was invented/discovered. But it ain't here yet, sister.

Personally, I've had about a 100 x-rays. Maybe I'll find out down the line that it did some damage. But without them, I wouldn't have lived past 21 y.o. and I certainly wouldn't be walking today. They have actually given me much more life than I would have had otherwise. It's a big reason why I became an x-ray tech. The alternative is a guy with a knife digging around in the dark trying to figure out why you're sick. (the good old days) No thanks!!

In the end, do what you feel is right for you. But when you make your decision, make a concerted effort to get the correct information. Newspapers/tv are the last place I go when I actually want to learn something of value. Medical journals can be found in any library and are available online as well. Remember, informed consent implies that the information used to come to the decision was thought to be as cpmplete/factual as possible.

**True story: In the spring, while I was a student, one of my x-ray mentors (and also a mammo tech and friend) had a lady from the ER who was having chest pain and a chest x-ray was ordered. As she was interviewing her to figure out what exactly was wrong, she burst out crying. Once she calmed her down, the tech finds out she is bleeding from her chest and that she has bandages taped on. Then she goes on to tell her that she's been bleeding under her left breast for months. She was hiding it because she was worried that her boyfriend would dump her if he knew. She never had it checked, never had a mammo performed. It was total carnage under there, just horrible. She was in end-stage breast cancer. It did not turn out happily.
** A good friend of mine happens to be a mammo tech and just had her 2nd mastectomy. Both times it was caught early while very treatable. She's one of the funniest, warmest people in my life and without those mammos, I would never have gotten a chance to meet her. (Love ya Steph!)

Good luck.

-X.