A cheap, comparatively simple and effective method is proposed for the large quantity production of the sheets of graphenic carbon materials (GCM) by annealing the mixture of nickel powder with a suitable carbon amount at the temperatures close to 1000 C. The number of graphene layers in the sheets of GCM may be varied by altering the amount of carbon in the mixture and parameters of annealing and drying of the obtained products. Samples of GCM were prepared in the form of heat-dried GCM paper and in the form of graphene sponge with freeze-drying. The appearance of GCM on the surface of Ni particles was identified using a scanning electron microscope (SEM) at a low accelerating voltage of 5 kV. The thickness and properties of the layers were investigated by electron microscopy and X-ray diffraction. The fabrication processes were carried out at the concentrations of added carbon from 0 to 1 at%. The results obtained are fully consistent with the well-known solid phase reactions of carbon dissolution in Ni at 1000 °C and graphene or graphite precipitation on the surface with cooling down to the room temperatures.