As I think even photons are absorbed by a non-transparent material, they aren't absorbed exactly at the same time. Light does propagate though a (sufficiently thin) opaque material. So can we measure the speed of light in such a material?
Answer
The speed of light in a medium depends on the frequency of the electromagnetic radiation $$v(\nu) = \frac{c}{n(\nu)}$$ where $n(\nu)$ is the refractive index. In a general case, $n(\nu)$ is a complex number, and its imaginary part accounts for the absorption of the medium (i.e. if a material is not transparent at frequency $\nu$, then $\textrm{Im } n(\nu) \neq 0$).
That being said, I can think of (at least) two ways to experimentally measure the speed of light in a medium at a given frequency.
The first one is a bit indirect. The speed of light can be calculated from the magnetic permeability $\mu$ and electric permittivity $\epsilon$ $$v = \frac{1}{\sqrt{\epsilon \mu}}$$ Most materials are non-magnetic ($\mu=\mu_0$), so you only have to measure $\epsilon$. This can be done, for example, using the material as a dielectric for a capacitor.
The other method is somewhat more "optical": when a light ray reaches a surface, part of its energy gets reflected and other part enters the material (and if the material is nontransparent, then it is completely absorbed). The amount of energy reflected and transmitted is given by the Fresnel equations, and it depends on the refractive index of the material (and on the angle of incidence and polarization as well). So you can compute the speed of light just measuring the intensity of light before and after being reflected.
Even better, if you use polarized light whose polarization lies in the plane of the ray and the surface normal. Then, at a special angle of incidence called Brewster's angle there is no reflection at all. The Brewster's angle (if the ray comes from air) is $$\theta_B = \arctan n$$
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