Is it unwelcome to pick a ‘Japanese name?”

I've seen in many places online that it's considered unwelcome/improper to choose a 'more Japanese' name. I've been studying the language for a while and feel like my name (which is Italian-American) doesn't sound good or flow when written or said in Japanese.

Is it true that it's not right to do this?

If not, does anyone know how I can go about trying to pick a Japanese name for when I'm speaking with friends (in America) or when I travel to Japan eventually?

I don't mean to be disrespectful in asking this, I'm just curious. Thanks!