Why can we produce a sound by blowing across a blade of grass, a paper strip or similar I just don't know how is the sound produced?
Does it come from the vibration of the blade of grass or the air in the up and down? 
 A: Vortex shedding. Blowing on the sharp edge causes separation of the flow stream and the creation of vortices; rotating pockets of gas. The vortices are periodic and impart energy to the blade of grass causing it to vibrate.
Vortex shedding is problematic in aircraft as it steals lift from the wing. But it is a useful as a flow sensing (measurement) technology - and making buzzing sounds with grass blades!
A: Vortex shedding is a good suggestion, but I suspect the cause might actually be aeroelastic flutter. It's a form of fluid/structure interaction involving a positive feedback instability between the angle of attack of an airfoil (e.g. the blade of grass) and the stiffness of the wing structure.  Basically, the lift on the airfoil causes deflection in the structure, which increases the angle of attack of the airfoil, which increases the lateral lift force, etc.  It ultimately induces a vibration in the structure, which may explain the noise produced by the blade of grass.
There is a video on YouTube here that shows examples of aeroelastic flutter in an airplane tail and the famous example of the Tacoma Narrows Bridge collapse (always a fun video to watch!).  I'll dig around and see if I can find a better video or image of how the phenomenon works ...
A: I am pretty sure the underlying phenomenon is indeed flutter. A taut flexible uniform band (such as a suspension bridge or a stretched blade of grass) will have a slightly higher frequency for pitching motions than for heave motions. This is characteristic of flutter. It would be nice to have someone high-speed video the moving grass blade for confirmation, though.
