I'm playing around with the idea of building an all out neutral deck, but a voice in the back of my head keeps telling me it's a bad idea.  Does anyone have any suggestions on how to make a sturdy deck with this concept?  Is it even allowed to use a non-faction deck?

It's absolutely allowed.

Personally, I'm not sure how you would build such a deck, but I remember reading on these forums that someone else has an all neutral deck (it might have been TheProfessor).  Regardless of how well it works, I can't see why you shouldn't give it a shot - sounds like a fun challenge!

 Great, thanks for the moral support.

 yes, I've given it a go, and had fun with it!   Of course if anyone plays Interrogation Center, then the deck loses! 

It's theoretically possible, but I don't know if it would work very well and there are some cards that punish neutrals which would make life very hard.

It is also a deck idea I had, but my main concern with it is that there is no real theme or game mechanic in the Neutral. Rather they are extras which can help another faction's basic mechanic or theme. So a mono-neutral deck would be a pack of interesting cards taken alone, but which wold not work together to make the break. And I don't think there is any other neutral card than Political Demonstration which can manage opponent's support, which is a must-have nowadays (apart from Azathoth, and Dimensional Rift of course).

The neutral cards definitly have their fare share of strong cards, as long as you have enough of them i think it will be easier than you think, i've turned the tide of many a game with neutrals, especially dimensional rift, i hate and love that card.