I'll try to keep my list somewhat short.Ghost - "Elizabeth"Graveyard - "As the Years Pass by the Hours Bend"Dark Tranquillity - "Terminus (Where Death is Most Alive)"Björk - "It's Oh So Quiet"Sybreed - "Revive My Wounds"
There, five songs. Number one...I don't even know what to call the genre, but it's rock/metal I suppose, they kick ass.
The second song is an awesome heavy blues rock song, it's so kick ass, it's very mellow and I think you'll like it. I sure as hell do.
The third song is a melodic death metal song. Don't let the words "death metal" scare you, but as you said, you're not going to label anything. Give it a listen, I personally love it, and I hope it will change your view a little on growling. Note that it's melodic death metal, not death metal. The two genres are not really similar at all.
The fourth song is...well...one of the coolest songs I know of, and I heard of it through MTV a few years back. It's some sort of pop. I don't know what you'll think of it to be honest. It's a bit odd.
The fifth, and last song I'll post for now, is an industrial metal song. Now, when I heard someone say industrial metal before, I thought to myself...urgh, Rammstein (I think they are awful btw), but these guys are brilliant. Sybreed is a band that has made me go "Holy shit".
I hope you like the songs that I posted, and I'm glad you've decided to at least try and broaden your horizons a little. :)