As it is right now, Relics are not on the loot lists of any bosses due to the fact that only one class can use each type. Instead it is in Emblem vendors. A useable solution, but not a very interesting one. My suggestion is simple: give the classes that use Relics class-specific quests within the suitable raid tiers to unlock Relics.
Take, as an example, Death Knights. Let us assume that with patch 4.1 comes two raids: The Firelands and Deepholme (in this example, Deepholme is harder than The Firelands). The quest for Death Knight tanks to gain their tier 12-equivalent Sigil would be to kill the two first bosses in The Firelands and the first boss in Deepholme whilst doing some sort of amazing acrobatic trick involving Death Grip and jumping around. Or something along those lines, I don't have enough imagination to come up with something fun
This could also open up the possibility of 'Hard Mode Relics', where you have to do a similar, possibly slightly modified quest in the hard modes to gain a small upgrade on your Relic. And last, but not least, if you simply cannot complete the quest for some reason (broken space bar, broken Death Grip button, you name it) you can buy it for an obscene amount of emblems.
Thoughts? Suggestions? I thought this could be a fun way of gaining new sigils, all part of a system where you get to be a bit more 'improvisational' with your character.