Are helicopters aerodynamically stable?
It's complicated ;)

There are two types of stability; dynamic and static.

If an aircraft is disturbed by, say, a gust of wind, it will deviate from its attitude but then will immediately and without control inputs return to its original attitude.  This is positive static stability.  If it remains in the disturbed attitude unless corrected, it has neutral static stability.  If it continues to deviate, it has negative static stability.

If the aircraft returns to its original attitude, overshoots a bit, then corrects back, overshoots a little bit less and so on, eventually returning to its original attitude, then it has positive dynamic stability.  If these oscillations continue and do not "damp" down until the original attitude is regained, then it has neutral dynamic stability.  If the oscillations grow increasingly large, then it has negative dynamic stability.

For helicopters, I will ignore those with autopilots, hydraulic controls and stability augmentation systems.

Imagine hovering.  A gust of wind hits the helicopter.  The disk will "flap" away from the gust and the helicopter will move with the gust.  When the gust stops, the disk now experiences airflow from the opposite direction, since it is now moving sideways.  The disk will flap back the other way, returning to neutral but the fuselage acts as a pendulum hanging under the disk and its momentum will cause it to swing further out, tilting the disk more and causing the helicopter to move faster in the opposite direction to the gust than is required to return to the original attitude.  Eventually, the airflow now comes from the direction the helicopter is moving in so the disk flaps away, the fuselage swings too far and......  Unless corrected, it will continue to swing backwards and forwards, rolling, yawing and pitching.  It is statically stable (because it tends to return to the original attitude) but dynamically unstable since it continues to swing backwards and forwards from the original attitude.

In forward flight (and unless you want to know, I'll skip the details which include what direction the gust comes from, the tendency of the fuselage to weathercock and "blowback" since the details all about which way the disk flaps and which way the fuselage weathercocks and are a lot of words!), a helicopter is statically stable and dynamically stable in yaw and is statically stable and dynamically unstable in pitch and roll.

You can take your feet of the pedals and the aircraft will be stable in yaw.  You can take your hand off the collective (although you should never do this - if the engine quits, you need that lever down fast) and it will stay where it is, ignoring perhaps vibration making it eventually move up or down.  

You cannot take your hand off the cyclic since this is the control which determines pitch and roll and corrects for dynamic instability on a constant basis.

On a light helicopter, you would be out of control within a second or two.  On a heavy, maybe three or four seconds.

In summary, a helicopter is statically stable and dynamically unstable in all three axes in hover and forward flight except yaw in forward flight where it is dynamically stable.