The pillared-graphene architecture is a conceivable way of conjoining graphene nanoribbons and carbon nanotubes (CNTs) in nanoelectronics. Especially promising is its capability to dissipate thermal energy in thermal management applications. However, the thermal boundary resistance (Kapitza resistance) at the graphene nanoribbon-CNT interface is a phonon barricade and a bottleneck for efficacious heat extraction. Parallel to strain studies on thermal conductance, this work is a first report on the effects of mechanical strain on the interfacial phonon dynamics in the pillared-graphene nanostructure (PGN). Molecular dynamics simulations are employed to derive the changes in phononics as axial, torsional, and compound strains of various degrees are applied on the PGN. The pillar lattice structure behaves dissimilarly to the different types of strains. In-plane transverse optical mode softening as induced by torsional strain is more effective than LO softening (triggered by tension) in minimizing the thermal boundary resistance. Essentially, it is shown that there is a strong relationship between strained PGN pillar lattice structure, interfacial phononics, and thermal boundary resistance.