Carbon nanotubes (CNT) and graphene are considered as potential future candidates for many nano/microscale integrated devices due to their superior thermal properties. Both systems, however, exhibit significant anisotropy in their thermal conduction, limiting their performance as three-dimensional thermal transport materials. From thermal management perspective, one way to tailor this anisotropy is to consider designing alternative carbon-based architectures. This paper investigates the thermal transport in one such novel architecture-a pillared-graphene (PG) network nanostructure which combines graphene sheets and carbon nanotubes to create a three-dimensional network. Nonequilibrium molecular dynamics simulations have been carried out using the AIREBO potential to calculate the thermal conductivity of pillared-graphene structures along parallel (in-plane) as well as perpendicular (out-of-plane) directions with respect to the graphene plane. The resulting thermal conductivity values for PG systems are discussed and compared with simulated values for pure CNT and graphite. Our results show that in these PG structures, the thermal transport is governed by the minimum interpillar distance and the CNT-pillar length. This is primarily attributed to scattering of phonons occurring at the CNT-graphene junctions in these nanostructures. We foresee that such architecture could potentially be used as a template for designing future structurally stable microscale systems with tailorable in-plane and out-of-plane thermal transport.