I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded
in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem,
a claim repeated in this Wikipedia entry. I have been unsuccessful in finding a description
of such a mapping, or an image of what the embedding looks like.  I'd be grateful to any pointers on this topic.  Thanks!

Addendum.
It seems Benoît Kloeckner's answer below is definitive.  What I asked for 
apparently does not yet exist, but is 
"in process" and will soon be available through the work of the Hévéa project.

[23Apr2012] This is taken from the link in DamienC's comment and Benoît's update in the latter's answer below:
   

A group of french mathematicians and computer scientists are currently working on this. The project is named Hévéa, and has already produced a few images. Edit: a few images and the PNAS paper have been released, see http://hevea-project.fr/ENPageToreDossierDePresse.html

Just a few word to explain what I understood of their method (which is by using h-principle) from the few image I saw in preview. Start with a revolution torus. The meridians are cool, because they all have the same length, as expected from those of a flat torus. But the parallels are totally uncool, because their lengths differ greatly: they witness the non-flatness of the revolution torus. 

Now perturb your torus by adding waves in the direction of the meridians (like an accordion), with large amplitude on the inside and small amplitude on the outside. If you design this perturbation well, you can manage so that the parallels now all have the same length. Of course, the perturbed meridian have now varying lengths! So you do the same thing by adding small waves in another direction, getting all meridians to have the same length again. You can iterate this procedure in a way so that the embedding converges in the $C^1$ topology to a flat embedded torus. But to prove that the precise perturbation you chose in order to get a nice image does converge, and that your maps are embeddings needs work (getting an immersion is easier if I remember well).

Also, the Hévéa project plans to draw images of Nash spheres, that is $C^1$ isometric embeddings of spheres of radius $>1$ inside a ball of unit radius.

Edit: Roman Kogan mentions in an answer below the following relevant, more recent links:


detailed paper on the isometric embedding of a flat torus including photographs of a 3D-printed model on p.67. 
video showcasing the 3D model. 

