I was born in a country that (somewhat) supported terrorism - Czechoslovakia. Eventually we switched sides (and later also somewhat supported terrorism again). Why did we switch sides? Because our emigrants received support from the West (while we were receiving very little terrorism). The West didn't engage the Warsaw pact directly (or semi-directly through terrorism), but through cultural war. That's why (at least I think) the West was successful in the Eastern Europe, but not in the Latin America, or in the Middle East.
In my opinion, immigrants are the key to successful foreign policy. The reason they are coming is because you're doing something better than their home leaders. And they will tell their relatives in their home country that, very passionately.
The terrorism is short-term strategy of states or other groups that are massively losing the cultural war (kinda like firing employees to boost profits). The long-term strategy is outlined above (and as every long-term strategy it entails element of getting better). So don't worry about the culture; if your opponent resorts to terrorism, your culture has already won!