Joseph, maybe among the Yiddish speaking chasidim, they use Yiddish names. But Litvish chareidim do not. If they name after someone with a Yiddish name, they change it to the Hebrew equivalent (eg Faiga becomes Tzipora, Gittel, Tova). I am speaking from my experience- from hearing the names in the playground, seeing the names on class lists, knowing girls who if they do have a Yiddish name (because they have American parents), hide them, change them, use only a middle/first name instead.
Back to the OP, most Americans who hear Sivan would think, why would someone name their kid after a month? If it was just a made up name with no other meaning, I think it would be less likely that the kid would feel uncomfortable.