Here is one that anime fans will recognize instantly, and that is Yoko Kanno, though she has composed and performed music for movies, television, video games and commercials.. She is what one might call, prolific, for her music cuts across a multitude of different genres. The music ranges from jazz, pop, rock, electronica, and more, while also every once and a while mixing different genres to great effect.
Honestly, there is not much to be said, other than to find her music and listen to it. To start off I will suggest that you listen to anything that was from the anime Cowboy Bebop (should watch that too, if you haven't). While most of the music there is mostly of the jazz genre, it is amazing nonetheless. Two songs that I will suggest from that is "Call Me" and "Gotta Knock a little harder" I think they will give you a good idea of what kind of sound they are going for. Another soundtrack I can recommend is for Escaflowne (either the series or movie, both have amazing music) the song "Escaflowne" is the perfect song to give you an idea of her orchestral music that has a powerful and bombastic delivery. Last suggestion is Ghost in the shell, the music here ranges from slow paced pop pieces, full rock n' roll, and heavy hitting electronica. Two songs that I feel are a must are "I do" and "Velveteen" two songs on opposite sides of the spectrum, but amazing on their own.
One of the great things about her music, is that one genre is not favored over another, the music she composes is created to fit the mood, characters, story, and motivations of what she is working on at that time. If you look at what she is composing for, and what music it creates proves that great care is made to create a mesh of music and subject matter that is something amazing, and improves the experience of whatever she composes for. She has worked with, a number of artists, the most notable being Maaya Sakamoto, who through the years has worked with her on many different projects. So I can say that if you have not heard of her yet, I have a feeling that she will have at least one song that you will enjoy.