Identify which face of a cube is up
A gyro won't tell you anything useful, at least not the kind of electronic gyro you can afford.  It sounds like what you want is an accelerometer. One for each axis lets you find the complete acceleration vector, which will be just gravity when the cube is still.  There are units with three accelerometers, each orthogonal, integrated into one device.  Search for "3-axis accelerometer".

If the cube is being jostled about, the acceleration vector won't point exactly up, but mostly it will.  Unless someone is deliberately whacking this thing or tossing it around, the acceleration vector will point upwards well enough.  You can even detect the in-flight case, since the acceleration vector will be zero then.