A turbine engine system includes a turbine engine shaft, a first rotor, a second rotor and an engine monitoring system. The shaft includes a shaft bore formed by a shaft wall. The shaft bore extends along an axial centerline through the shaft between a first shaft end and a second shaft end. The first rotor is connected to the shaft at the first shaft end, and the second rotor is connected to the shaft at the second shaft end. The engine monitoring system includes a sensor connected to the second rotor, a transmitter arranged at the first shaft end, and a plurality of conduit assemblies. Each conduit assembly includes a conduit that extends axially within the shaft bore and is connected to the shaft wall. A first of the conduit assemblies also includes a wire that extends through a respective conduit and connects the sensor and the transmitter.