According to this article: http://physics.aps.org/articles/v5/24:
The statement that in graphene the "conduction electrons are massless" is because the energy levels (bands) are proportional to their momenta.
So the $E = \sqrt{p^2+m^2}$ relation of a free electron becomes $E\propto p$ in graphene.
Massless particles travel all at the same speed because of the $E\propto p$ relation but this characteristic velocity in graphene is far below c though, only 0.3% of the speed of light.
The reason that the relation $E\propto p$ leads to a characteristic speed is due to the quantum mechanical wave character. $E$ is proportional to the phase changes in time, $p$ is proportional to the phase changes in space and therefor $p/E$ is proportional to the velocity. In the case that $E\propto p$ there is a characteristic velocity $v$ independent of the energy level.
The most striking aspect of graphene is that its electronic energy
levels, or “bands,” produce conduction electrons whose energies are
directly proportional to their momentum. This is the energy-momentum
relationship exhibited by photons, which are massless particles of
light. Electrons and other particles of matter normally have energies
that depend on the square of their momentum.
When the bands are plotted in three dimensions, the photonlike
energy-momentum relationship appears as an inverted cone, called a
Dirac cone. This unusual relationship causes conduction electrons to
behave as though they were massless, like photons, so that all of them
travel at roughly the same speed (about 0.3 percent of the speed of
light). This uniformity leads to a conductivity greater than copper.
Hans
