If you look at the Wikipedia link tokenadult posted, it's clear that there are huge variances both over time (http://upload.wikimedia.org/wikipedia/commons/7/72/Hungarian...) and between different countries (http://en.wikipedia.org/wiki/List_of_countries_by_suicide_ra...).
That strongly suggests that some component of a society does influence the suicide rate. Of course, it's not the only factor, but it is surely a factor.