Planar graph via the intersection of fat thingies? There is a beautiful theorem of Koebe (see here) that states that any planar graph can be drawn as kissing graph of disks (very romantic...). (Putting it somewhat differently, any planar graph can be drawn as the intersection graph of disks.) 

Koebe theorem is not very easy to prove. My question: Is there an easier version of this theorem where instead of disks one is allowed to use any fat convex shapes (convexity might be open to negotiations, but not fatness). Note, that every vertex can be a different shape.

Thanks...

Clarification: For a shape $X$, let $R(X)$ be the radius of the smallest enclosing ball of $X$, and let $r(X)$ let me the radius of the largest enclosed ball in $S$. The shape $S$ is $\alpha$-fat if $R(x) /r(x) \leq \alpha$. (This is not the only definition for fatness, BTW.)
You didn't say the fat objects had to be two-dimensional, did you? Felsner and Francis prove that it's always possible with axis-parallel cubes in 3d. But, the proof involves Schramm's generalizations of Koebe-Thurston-Andreev, so it's not exactly a simpler result. They also mention along the way that for four-connected maximal planar graphs it's possible to use parallel-sided equilateral triangles.