In this study, the effects of a hybrid filler composed of zero-dimensional spherical AlN particles and two-dimensional BN flakes on the thermal conductivity of epoxy resin were studied. The thermal conductivity (TC) of the pristine epoxy matrix (EP) was 0.22 W/(m K), while the composite showed the TC of 10.18 W/(m K) at the 75 wt% AlN–BN hybrid filler loading, which is approximately a 46-fold increase. Moreover, various essential application properties were examined, such as the viscosity, cooling rate, coefficient of thermal expansion (CTE), morphology, and electrical properties. In particular, the AlN–BN/EP composite showed higher thermal stability and lower CTE (22.56 ppm/°C) than pure epoxy. Overall, the demonstrated outstanding thermal performance is appropriate for the production of electronic packaging materials, including next-generation flip-chip underfills.