Dielectric polymer composites with high thermal conductivity are very promising for microelectronic packaging and thermal management application in new energy systems such as solar cells and light emitting diodes (LEDs). However, a well-known paradox is that conventional composites with high thermal conductivity usually suffer from the high dielectric constant and high dielectric loss, while on the other hand, composite materials with excellent dielectric properties usually possess low thermal conductivity. In this work, an ideal dielectric thermally conductive epoxy nanocomposite is successfully fabricated using polyhedral oligosilsesquioxane (POSS) functionalized boron nitride nanotubes (BNNTs) as fillers. The nanocomposites with 30 wt% fraction of POSS modified BNNTs exhibit much lower dielectric constant, dielectric loss tangent, and coefficient of thermal expansion in comparison with the pure epoxy resin. As an example, below 100 Hz, the dielectric loss of the nanocomposites with 20 and 30 wt% BNNTs is reduced by one order of magnitude in comparison with the pure epoxy resin. Moreover, the nanocomposites show a dramatic thermal conductivity enhancement of 1360% in comparison with the pristine epoxy resin at a BNNT loading fraction of 30 wt%. The merits of the designed composites are suggested to originate from the excellent intrinsic properties of embedded BNNTs, effective surface modification by POSS molecules, and carefully developed composite preparation methods.