Dear Mauro,
sorry for the delay, sometimes it's difficult when one has a day job, but I suppose you know that ;)
So about proving anisotropy of propagation speed in a regular lattice: The notion of Minkowski sum I mentioned is indeed the one you are familiar with. The anisotropy proof goes as follows: think of the lattice as projected onto space, ignoring the time dimension. Designate a certain starting point as the origin. Then define the "ball" B_n to be the set of points which can be reached in n steps from the origin. Clearly, every B_n is a polytope, i.e. is the convex hull of finitely many points. For a certain n (n=2 in your case), the extreme points of B_n are all translates of the origin. Then how far can we get in n+n=2n steps? From the origin, we can get to all the outer points of B_n; from each outer point, we can then traverse another n steps. And then the distance traversable in these n steps is precisely given by a translated copy of B_n! Therefore, B_2n is the Minkowski sum of B_n with itself. Hence B_2n coincides with B_n scaled by a factor of 2. The same argument applies inductively to show that
[math]B_{kn}=k\cdot B_n[/math]
In particular, the shape of the balls B_kn is independent of k.
Concerning the comparison to graphene, yes, that's an excellent question! One difference is that we are now looking at wave functions instead of classical point particles. Then the characteristic quantity of the system is the energy-momentum relation E(p) of the (quasi-)particles. A Taylor expansion of this quantity yields precisely something of the form
[math]E(p)=M_{ij}p^i p^j + O(p^3)[/math]
where M_ij is something like an "inverse mass tensor" and summation is implied. When
[math]M_{ij}=\delta_{ij}[/math]
holds, then the low-energy excitations have isotropic propagation speed! And as I mentioned in my essay, it is in fact only the low-energy excitations for which the whole emergence of the massless Dirac equation holds. (In light of this discussion, this is a point which I should have emphasized more...) For higher-energy excitations, isotropy does not hold. In the graphene case, anisotropies occur which are known as "trigonal warping"; I haven't been able to find a good reference for this, but google turns up a whole lot of papers on that.