In America we also would say something similar: "it doesn't really do anything for me" or "I'm not really into it." You can use those phrases without "really" but I think a lot of Americans, myself included, tend to use "really" in those phrases. Our version of "cup of tea" would be "it's not really my thing." Those are polite ways and relatively neutral ways to say something doesn't interest you. If you wanted to word it more strongly and sound more opinionated you could say "I couldn't care less". For example: "I couldn't care less about her new boyfriend" or "I couldn't care less what he thought about me." It implies that you can't even be bothered to think about that person/topic/thing. It often has a feeling of being dismissive or a little dramatic. So, it's probably not a phrase you'd use with a professor or police officer, but you could use more freely with friends. Be warned, a lot of Americans butcher this phrase and say "I could care less." That's not right :)

In England we might say "It doesn't do it for me" or "I'm not into it" which would mean "It doesn't interest me" or "I don't really like it" OR we could also say "It's not really my cup of tea" ;)

all these options sound unnatural. If I was put in the music situation I would reply with, " I don't really like this kind of music, but i like ______ kind of music." you could also say, "I don't care", but be careful because this can come across as rude.