PNG alpha rendered as black in Blender

Question

I'm a Blender novice, so this is probably easy to fix.

When I use a transparent PNG as a texture in Blender, the parts that should be transparent are rendered as black.

This is especially confusing since in the material preview it looks as if the material would indeed be transparent.

Here's a screenshot:

This is the test texture, and in the right on top of a checkerboard:

                      

Here is the .blend file in case you want to check it:

                                                     

Edit: After playing with the settings, it does render with transparency in the rendered output, but that's not what I need. I want to use the models with Three.js, so I want to just quickly see how it looks in Blender, I don't need more than simple "Z-transparency" ("regular" one, non-raytraced) here's a reference rendering of what I expect to see in Blender while I edit (this is the same model rendered with Three.js on an HTML <canvas>):

                                               

Answer 1

Just in case none of the two nice fellow at BlenderArtists chose to answer here, here's the two possible ways to do it:

SxJP's answer:

In the viewport, click N, open the "Display" tab and under "Shading", select "GLSL".
Click the link above for the original post.

                                    

Sanctuary's answer:

This one allows you not having to use the GLSL renderer. To avoid duplication, click the link to go to the answer, it has handy screenshots.