Recently, metal-based composites reinforced with carbon nanofillers such as graphene sheets (SLGSs) and carbon nanotubes (CNTs) were proposed to combine the properties of metals. These nanofillers have led to novel materials for a variety of applications. Although there are various experimental and numerical studies in this regard, there is a lack of fundamental understanding of how the geometrical characteristics of these different nanofillers affect their mechanical properties. Here, we report a series of full atomistic molecular dynamics (MD) simulations aimed to assess the issue. Two distinct computational cells, SLGS/Cu and CNT/Cu samples, that have copper as the matrix material, are examined under identical conditions to explore the role of nanofiller geometry on the elastic constants of these nanocomposites. Moreover, the temperature dependence of the axial Young's modulus for each case is investigated via MD simulations. Finally, we also studied the effect of graphene/matrix interfacial interactions on the elastic constants of metal nanocomposites by comparing the results of the SLGS/Cu case with the ones determined for a similar SLGS/Ni sample. Our results reveal that graphene should be the nanomaterial of choice to strengthen metal matrices. Keywords: Metal matrix nanocomposite, Molecular dynamics method, Young's modulus, Graphene sheet, Carbon nanotube