I guess it must be in bad taste to answer my own question, but I now know a bit more about this problem and maybe there are other people interested in references to it.
As I mentioned in the question, Ehrhart himself settled the $2$-dimensional case and the case where the convex body is a simplex. Apparently nothing interesting was done until late last year when Robert J. Berman and Bo Berndtsson http://arxiv.org/abs/1112.4445 used PDE techniques to settle the case where the convex body is dual to a Fano polytope.
Reminder. A Fano polytope is a lattice polytope such that the vertices of each facet define a basis of the lattice.
As for the weak version of the Ehrhart conjecture: it is true. The preprint will be soon on the ArXiv.
Addendum. Here is the proof of the weak (or asymptotic) version of the Ehrhart conjecture modulo the following result that will appear in a forthcoming paper by Balacheff, Tzanev, and myself.
Theorem (ABT). If the origin is the unique integer point in the interior of a convex body $K \subset \mathbb{R}^n$, then the volume of the dual body $K^*$ is at least $(\pi/8)^n (n+1)/n!$.
The conjecture is that the volume of $K^*$ is at least $(n+1)/n!$, but we were able to show this only for $n=2$.
Now for the weak version of the Ehrhart conjecture: let $K \subset \mathbb{R}^n$ be a convex body with its barycenter at the origin and containing no other integer point. By the Blaschke-Santalo inequality (actually a version of it that holds for asymmetric bodies and their duals with respect to the barycenter, but I don't remember to whom it is due!!), we have
that $\epsilon_n^2 \geq |K||K^*|$, where $\epsilon_n$ is the volume of the Euclidean unit ball.
This inequality, together with the ABT theorem yields
$$
\epsilon_n^2 \geq |K||K^*| \geq |K| (\pi/8)^n (n+1)/n!
$$
or $|K| \leq (8/\pi)^n\epsilon_n^2 n!/(n+1)$.
Now, the proof reduces to showing that the quantity
$$
\left(\epsilon_n^2 (n!)^2/(n+1)^{(n+1)}\right)^{1/n}
$$
is bounded above by a quantity $C$. This is just the $n$th root of the quotient of the volume product of the ball and the volume product of the simplex and, therefore, standard fare in asymptotic geometry. This quantity is asymptotically $2\pi / e$ and seems to be always less than $3$. I'll check and edit when I have a bit more time.
I worked out this proof at the early stages of my collaboration with Balacheff (before Tzanev joined us), but this result got dropped out of the (forthcoming) paper. I reproduce it here from my notes, but I'm a bit rusty on the details. Very possibly, one can bypass ABT and apply a judicious mixture of Minkowki's lattice point theorem, Rogers-Shephard inequality, the Bourgain-Milman theorem, and the asymmetric version of Blaschke_Santalo to get the proof. In any case, those are the basic ingredients that go in the proof above.
