Diamond

Actually diamonds have a high refractive index of 2.42 ; because of which critical angle for diamond-air interface is very small, about 24.4^o .Now the diamond is cut in such a fashion that most of the rays entering the diamond suffer multiple internal reflections until they reach the top face of the diamond where they exit leaving the diamond sparkling.

In case of ordinary glass piece with diamond cut , the critical angle has greater value of about 42^o

so the light rays entering it doesn't suffer total internal reflection ( angle of incidence being les than the critical angle) ; hence it doesn't shine.