How do modern avionics work without vacuum pumps or gyroscopes?
If you haven't already done so, you should have a look at section 7 of the C162 POH. That section gives an overview of all the aircraft's systems and the information is at a good level of detail for a checkride (at least at the private level). The examiner isn't expecting you to explain how a ring laser gyro works, but he probably will expect you to be able to give an explanation like this (from the POH):


  AIR DATA, ATTITUDE AND HEADING REFERENCE SYSTEM (ADAHRS) AND MAGNETOMETER (GRS)  
  
  The ADAHRS provides airplane attitude and flight characteristics
  information to the G300 displays integrated avionics units. The ADAHRS
  unit, located behind the instrument panel, contains accelerometers,
  tilt sensors and rate sensors that replace spinning mass gyros used in
  other airplanes. The magnetometer, located in the tailcone, interfaces
  with the ADAHRS to provide heading information.
  
  The air data portion of the ADAHARS compiles information from the
  airplane's pitot-static system to calculate pressure altitude,
  airspeed, true airspeed, vertical speed and outside air temperature.
  An outside air temperature probe, mounted on top of the cabin, is
  connected to the ADAHRS.


The G300 Pilot's Guide goes into more detail about the system's components (see Section 1), although it's still at a fairly high level.