Alice in Chains (yes I know they're classified as grunge but they've always had a more metal sound)
I'm tryin to come up with ones that probably haven't been said, don't do a lot of screaming, and have "more musical" music as opposed to raging guitars playing the same note over and over again.
Iron Maiden - They've been said many times already but they're just amazing. Avenged Sevenfold - They don't do screaming at all and have really nice guitar solos that are cleaner than many other guitar solos in modern metal. System of a Down - Some of their stuff is not very melodic and is that whole repetitive heavy guitar stuff but some of their stuff is really interesting. Breaking Benjamin - They're not as metal as most of the other bands I listed but their stuff is more melodic. Queensrӱche - They sound somewhat similar to Iron Maiden. Nonpoint - They're one of my favorites. Deftones - Their music is very mellow. Disturbed - Most of their songs are so similar it becomes repetitive but he sings more than he screams. Skindred - They're just very fun to listen to because of their lead singer's accent and their fusion of metal and stuff like dubstep and reggae. Babymetal - They're music is a fusion of heavy metal and j-pop so there's no screaming and it's just interesting to listen to them. Black Veil Brides - They're melodic.