How to find what are the dominant safety concerns a tourist should have in a country?

You can have a look at the travel advice that many countries offer:

http://www.fco.gov.uk/en/travel-and-living-abroad/travel-advice-by-country/

http://www.voyage.gc.ca/countries_pays/updates_mise-a-jour-eng.asp

http://www.diplomatie.gouv.fr/fr/conseils-aux-voyageurs_909/index.html

http://travel.state.gov/travel/cis_pa_tw/cis/cis_4965.html

http://www.smartraveller.gov.au/zw-cgi/view/Advice/Index

etc ...

These sites give you very detailed and up-to-date information on the topics you have mentioned.

Health and safety concerns are also covered by any good travel guidebook. It is clear that this information might not be up-to-date. You can also find some hints in the country fiches published on the websites of these travel guidebooks.

I'm almost always check the websites of the ministries of foreign affairs of my neighboring countries.

The reason that I choose neighboring countries is easy: my own country is too small to have such information. And I think neighboring countries have more or less the same safety standard and culture so that their evaluation of a safety situation in other countries will fit to my personal estimation.

For example this is the site of Switzerland (unfortunately only in German, French and Italian) but I think you will get a feeling.

This site is also very useful. It is in German, but it is obvious how to use it. Just choose a country on the Map and then you will get a link collection for this country. A lot of these links are then available in English. An advantage of this site is that you'll see the general situation of the whole world in an overview.

If you want the American point of view of each country then the State Department is the place to go.

They do a profile on each country that seemingly has the information you are looking for.

If you are travelling with a printed travel guide book they should all have a relevant section to address your concerns.

For instance in Lonely Planet travel guides there will be a section called Dangers and annoyances.

For Australian citizens there are "travel advisories" issued by the government. They seem to change their name from time to time for no good reason. Right now it's called "Smartraveller" and has a website at www.smartraveller.gov.au, in my experience though they are pretty good at making just about any potential destination seem scary. I'm sure they could issue a warning about Australia if they felt like it saying "Many dangerous animals, snakes, and spiders" for instance (-: