Wireless fully-passive neural recording systems are demonstrated with heterogeneous integration of sensing, mixing and communication components in flexible polymer dielectric films. The recording neurosensor has an antenna, antiparallel diode, and a by-pass capacitor to modulate an incoming carrier signal with the neuropotentials and backscatter the mixed signal to an external reader circuit. Planar antennas are realized on Liquid Crystal Polymer (LCP) flexible substrates with low dielectric constant. The designed antenna topology is realized with a single metal layer and eliminates the dependency on substrate thickness and leads to thinner sensors. This is a major advance compared to prior passive neural recording with rigid and thicker substrates and components. Approximately 80% thickness reduction and 20% volume reduction are achieved with similar performance compared to earlier studies. Our approach thus leads to low-cost and disposable wireless neurosensors on a flexible platform with heterogenous integration.