Why does the Airbus A320 series seem to rock left and right in level flight?
As nobody has answered this in 4 days I'll add this "layman's" input.

I am not a pilot or aeronautically trained, but I understand that essentially all airliners have a dynamic yaw instability aka Dutch Roll and have stabilisers designed to address this. ANY feedback error correction system requires SOME error signal to derive the feedback from and how much error and how it manifests changes with system. 

@mins I'm no expert in this area BUT all aircraft are different and you'd be surprised (or should be) if they all behaved the same. The overall combination of a wide range of dynamic forces leads to the end result. 

As I recall (probably from Peter R March's very interesting 'The 747 story'  Dutch roll was not a major factor in large aircraft until the larger jets and they lost a prototype with an experienced pilot inexplicably. Subsequent investigations and close calls showed the cause was a dynamic yaw-roll coupling. AFAIR this is caused by the high yaw-induced rolling moment of swept wings and as these are standard fare as speeds rise all large/fast modern airliners have this issue and have a yaw damper to make it unnecessary for the pilot to deal with it.
Without a yaw damper, the vertical tail would need to be much larger to provide enough damping. This costs weight and drag and creates problems of its own due to the location of the tail upwards of the line of symmetry. When the aircraft yaws, the wing creates a rolling moment. This rolling motion causes a drag difference which, together with the directional stability of the tail, swings the aircraft's tail around to the other side, and the process repeats in the opposite direction. 

It may be, and I'm 'mentating' this with no proof at all, that they are up against a regulatory barrier and that correcting the problem may lead to them having to do something else expensive. eg early British cars were long stroke not for any technical reason but due to taxation advantages as taxing was based on bore AREA. For a given cc size bore area was reduced by increasing stroke. This limited achievable RPM. Japanese engines had no such limits and I remember our amazement in the early days of Japanese motorcycles at the under-square high revving Japanese motorcycles - which gave them a power per cc rating advantage. 

Similarly some America's cup yachts at one stage had their ratings improved by tacking light metal sheeting on their decks. This made NO difference whatsoever.

As an example of stabiliser interaction in another field - many modern cameras have inbuilt motion stabilisers. These need to be turned off when the camera is used on a tripod - or else the compensation system causes the camera to "hunt" about the essentially perfectly stable tripod. I've read of people who had major blur issues through failing to realise this.