"between" really only applies to two places at a time, while "among" includes three or more, so it all depends on how many countries you're talking about. One does not travel "among the two countries" nor "between the three countries". I suppose if you grouped countries, you could use "between" for more than two, as in "He travels between the US and Sweeden, Norway and Denmark."
It's as good as the (English) translation here. Who knows what it takes to get from one town to another, but no one I know ever called that "traveling." "They go from one town to another," "They are going from one town to another." But, in English, unless only two towns are specified, "among" is the acceptable translation, and not "between." Between is for two things....
attraverso is probably better Italian.
You have to be careful with what "through" means in English, as in "he fell through the ice", "he succeeded through hard work" "He is through with his job" - some translations of "tra" = "through" might work, but traveling "through" countries probably isn't what tra means.