I'm not familiar with Flash, but as I understand it I don't see how that would work since I need 3D software to move the object in 3D space.
I would just do something like this in flash. If I was going to do it in maya I would just make a plane and use trans maps to create everything...I don't know if this would actually work but that's the first thing I would try.
You're correct, it was a drawn animation in the video. However, I want to achieve the same visual with 3D software because I need to replicate the concept for many 2D images.
I highly doubt that this animation was done in any 3D program it is simply cleaver drawing. I think what you are really asking about turning a drawing into a 3d mesh would be better achieved in Zbrush rather than Maya. Google Zsketch that might give you some good ideas.
You mean do this in Maya? How?
Depending on what you are going to use it for I suppose you could use a few simple cubes or planes with some fancy transparency effects to give the appearance of 3d that could probably recreate your little video there easily enough.