Updating this post to help anyone trying to unlock the colors for Might of the Grizzlemaw.
You have to have completed Might of the Grizzlemaw from Legion.
The option to unlock the Brown skin “Complete 10 different Legion dungeons after unlocking a challenge appearance.”
The KB article https://us.battle.net/support/en/article/000105045
You are no longer required to use your current specialization’s hidden or challenge artifact appearances to gain progress towards unlocking its 3 color variants. You can now gain progress towards unlocking the 3 color variants of these appearance sets while you are in any spec and using any artifact appearance. While any artifact appearance may be used, you must have at least unlocked the base appearance for any given spec, and still complete these challenges using the same character who unlocked the base appearance.
The appearance requirement, “Defeat all Legion Dungeons”, has been changed to “Complete 10 Different Legion Dungeons”, and now includes the Seat of the Triumvirate dungeon.
This specifically means to complete 10 of any of the 12 dungeons listed below.
Arcway
Triumvirate
Maw of Souls
Cathedral
Halls of Valor
Neltharions Lair
Vault of the Wardens
Darkheart Thicket
Black Rook Hold
Eye of Aszhara
Assault on Violet Hold
Return to Karazhan
Running Maw of Souls Normal, Maw of Souls Heroic, Maw of Souls Mythic still counts as running 1 dungeon for this unlock.