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We present a reliable measurement process using a laser flash technique to measure intrinsic thermal diffusivity of vertically carbon nanotube (VACNT) arrays. The thermal diffusivity of VACNTs synthesized by a traditional water vapor-assisted thermal chemical vapor deposition process was measured, and shown to be ~30 mm2 s-1. The thermal conductivities of the VACNT film and the individual CNTs are around 27 and 540 W m-1 K-1, respectively. The effects of buckling, packing density, and intertube coupling on the intrinsic thermal diffusivity were examined quantitatively. Buckling decreases the thermal diffusivity dramatically; an increased packing density is beneficial in increasing the collective diffusivity, and therefore, the thermal conductivity of the VACNT film. By increasing the packing density, we achieved 70~90 W m-1 K-1 of VACNTs. It is the highly reliable measurement technique that renders us the feasibility of such systematic and quantitative parametric studies. Such systematic studies are highly important in providing researchers with a necessary vision on interpretation of the experimental data in literature, and a guideline to development of VACNT TIMs. The work also fundamentally provides the research community with a reliable technique for measuring thermal transport properties of various heterogeneous nanostructures.