With the rapid development of high-power-density electronic devices, interface thermal resistance has become a critical barrier for effective heat management in high-performance electronic products. Therefore, there is an urgent demand for advanced thermal interface materials (TIMs) with high cross-plane thermal conductivity and excellent compressibility to withstand increasingly complex operating conditions. To achieve this aim, a promising strategy involves vertically arranging highly thermoconductive graphene on polymers. However, with the currently available methods, achieving a balance between low interfacial thermal resistance, bidirectional high thermal conductivity, and large-scale production is challenging. Herein, we prepared a graphene framework with continuous filler structures in in-plane and cross-plane directions by bonding corrugated graphene to planar graphene paper. The interface interaction between the graphene paper framework and polymer matrix was enhanced via surface functionalization to reduce the interface thermal resistance. The resulting three-dimensional thermal framework endows the polymer composite material with a cross-plane thermal conductivity of 14.4 W·m−1·K−1 and in-plane thermal conductivity of 130 W·m−1·K−1 when the thermal filler loading is 10.1 wt%, with a thermal conductivity enhancement per 1 wt% filler loading of 831%, outperforming various graphene structures as fillers. Given its high thermal conductivity, low contact thermal resistance, and low compressive modulus, the developed highly thermoconductive composite material demonstrates superior performance in TIM testing compared with TFLEX-700, an advanced commercial TIM, effectively solving the interfacial heat transfer issues in electronic systems. This novel filler structure framework also provides a solution for achieving a balance between efficient thermal management and ease of processing.