Hi - that's a good question; lots of people worry about it.
To put your mind at rest:
Everyday, the your body actually gets hit by a certain amount of 'radiation' (the sort you are worrying about) from the sun. The dose is so small that most of us rightly go around unconcerned about it. (This does NOT apply when you go suntanning without sunscreen, though. This amount of exposure CAN be very harmful.)
Now, most X-rays only give off one day's worth of ambient sun radiation per X-ray. Put another way, it's like living another day in terms of radiation dose. So it's really quite safe. The radiation shields that radiologists and radiographers use are there because these people irradiate tens of people a day, so it adds up for them.
CT Scans (CAT scans) are quite a lot higher, but still nothing to be excessively worried about. The amount of harmful radiation given off depends very much on the length of body being imaged, and how much detail is required, but most CT scans give off about a year's worth of ambient sun radiation.
(For what it's worth, MRI scans don't give off any 'harmful' radiation of the sort that is concerning you. They operate on a different technique altogether.)
Lastly, it is obviously important to remember that the imaging studies are done for a reason, and so avoiding them unduly can often be a far greater risk to your health than getting hit by, say, a day's radiation. But you are quite right and justified in being concerned about it.