Adamantium Ore spawn points:
Norenen-dur, The Teeth that Gnash - Once you get to the point in the Tribunal Main quest, go through the following: Plaza Brindisi, Bamz-Amschend, Heartfire Hall, Passage of Whispers, Radac’s Forge, Passage of theWalker, Norenen-dur.
Old Mournhold, Armory Ruins - Go to the Great Bazaar then go through: Old Mournhold: Bazaar Sewers, Palace Sewers, West Sewers, Battlefield, City Gate
Old Mournhold, Temple Catacombs - Start in the Temple Basement then go through the Temple Sewers and then into the Temple Sewers East.
Old Mournhold, Temple Crypt - Start in the Temple Basement then go through the Temple Sewers, Temple Sewers East, Temple Gardens, Temple Shrine.
Give enough ore to Bols Indalen in the Craftsmen's hall and he'll make you all the pieces
(Source:
http://www.geocities.com/quanze_13)
It is easier if you follow Wraths tip, just visit
http://www.elderscrolls.com 
Other Adamantium locations:
- Apelles Matius in Ebonheart has a full set (but no helm) - Just make sure you talk with him before you kill him, otherwise you won't be able to finish Tribunals main quest.
- Yagak gro-Gluk in the Craftsmen's Hall (afaik also important for the main quest!) has the following pieces equiped: boots, cuirass, greaves, right bracer, right pauldron.
- You can find a pair of boots in the Bazaar sewers.
- In the Moril manor (reached trough the Mournhold sewers) you can find a pair of boots, a helm, a left bracer and a right one.
- A cuirass can be found in Temple Sewers West.