You are right, there is no safe level of radiation.
the smallest amount of radiation increases the risk of developing cancer in the future.
Still, I think there is hardly a reason to worry in your case.
Everybody is exposed to radiation as it is present in our atmosphere. It is called Natural Background and is about 3.1 mSv/year.
Now lets do statistic calculations:
A male of 27 years has a baseline risk of developing cancer of 44.9%.
A male of 27 years who has had 6 X-ray exposures (2 neck, 2 lower back and 2 upper back) has an increase in risk of 0.0514%, making the chance to develop cancer 44,95%.
A small increase in chance but it could have saved you a lot of trouble if they had found damages in your spine.
Dental X-rays are even much lower in dose. 1 dental X-ray increases your risk about 0.00004%
Requesting non radiation sources of imaging isn't wise. Those kinds of determinations should be left to physicians. Why? Because not all imaging modalities are the same, and some are used very specifically in certain instances because they provide the most information. Requesting an ultrasound over say, a CAT scan for any given procedure because you fear the radiation defeats the entire purpose of the exam, which is to give the physicians the most complete picture possible of a given situation, and some modalities simply are not suited for certain things, meaning having them done just because you want them to be can result in a serious waste of time and money.
The amount of radiation you receive is very minimal, stay in out of the Sun a little bit and you'll negate the effects if you are that worried. It really isn't something to worry about there are a number of more cancerous things in your life than a few necessary medical images taken I'm sure.