Power electronic devices and semiconductor packages generate high localized heat fluxes that must be diffused laterally using heat spreading materials to avoid hotspots. Engineered heat spreading materials like pyrolytic graphite and carbon-composite sheets offer higher in-plane thermal conductivity than conventional heat spreaders but exhibit anisotropic conduction behavior. Design of thermal management solutions is in part limited by the lack of accurate characterization techniques for such types of highly anisotropic materials. A popular technique for measuring thermal diffusivity is the Angstrom method, where a thin long specimen is heated periodically at the one end, and the corresponding transient temperature profile is measured at discrete locations along its length. The amplitude and phase of the temperature oscillations are then used to calculate thermal diffusivity. In its conventional form as described, this method is limited to characterization of specimens under 1D heat flow, which requires multiple measurements to be performed with differently prepared specimens to characterize anisotropic materials. We present a new measurement technique for characterizing the anisotropic in-plane thermal conductivity of thin films and sheets, which is an extension of the traditional Angstrom method. The measurement enhancements include non-contact 2D temperature mapping using infrared imaging and laser-based periodic heating. The specimen is suspended over an aperture and heated periodically at its center using a diode laser. The transient temperature distribution is measured on the opposite face of the specimen using an infrared microscope. Assuming a time-periodic solution for heat diffusion in a 2D domain with periodic heating, the corresponding amplitude and phase of the temperature oscillations are used to fit the thermal conductivities along the orthotropic directions using a numerical approach. To validate the accuracy of this technique for anisotropic materials, numerical models are developed to generate transient temperature profiles for hypothetical anisotropic materials of known properties. The resultant temperature profiles are processed through a fitting algorithm to extract the in-plane thermal conductivities, without the knowledge of the input properties to the model. These results agree well with the input values, which validates the accuracy of this measurement technique. Experiments have been conducted with a known isotropic reference material as well as an anisotropic heat spreading material, and the measured thermal conductivity values are consistent with the values reported in literature and data sheet. The limits of accuracy of this technique are being identified by validating it across a range of thermal conductivities and anisotropy ratios. This will enable standardization of this measurement technique, which would then aid the development and characterization of custom materials with desired anisotropic properties.