The ability of a ground effect (air)craft to able to fly with (small) perturbations in height from the ground is called height stability. In general, height stability is achieved if the derivative of the lift coefficient with increasing height is negative. In this case, a decrease in height (due to waves etc.) causes an increase in lift. As a result, the aircraft is returned to its original height due to the increased lift (with the opposite happening when height is increased).
As the ground effect increases as the height decreases, designing the ground effect aircraft with height stability is not very difficult and most of the ground effect aircraft designed and operated have height stability. A good example is the Lun Ekranoplan, which returned to its height after the height was altered by approximately 0.5m (due to missile launch).
"Lun Ekranoplan" by Soviet Navy - Stock Archives of Soviet Navy. Via Wikipedia.
Also, the Sea state limitations are a scale phenomenon — the larger the ground effect aircraft, rougher the seas it can handle. The Russians (Soviets) have the most experience in operating ground effect vehicles and according to them, sea state limit for a 500-tonne aircraft is around 2.5 m and the safe operating heights in terms of mean wave heights is given by,
$h = \frac{1.54 H_{\frac{1}{3}}}{2} + 0.1 c$
where,
$h$ is the vertical height measured from the mean wave height,
$c$ is the chord,
$H_{\frac{1}{3}}$ is the average of the 1/3 highest wave.
When this height exceeds the ground effect height the craft operates as an aircraft at a lower efficiency.
The main problems in case of operating a ground effect aircraft in choppy waters (as you noted, they are mostly operated in 'calm' seas) are twofold:
Takeoff and landing are dangerous in rough seas, due the impact loads on the hull and wings. This is the most important limitation for operating in rough seas.
Stability — The wing is already unstable, and near the ground in high lift region, it is more so; this is the reason most of the ground effect vehicles have proportionally larger tail surfaces compared to the conventional aircraft. A good example is the Soviet -90 'Orlyonko', which had tail surface nearly 50% of that of the wing.
"A-90 Orlyonok" by Kaboldy - Own work. Licensed under CC BY-SA 3.0 via Commons.
The main problem with waves (or in general, variations in height) is that the pitch stability changes with height. This results in the requirement of a large amount of control power in order to maintain trim — another reason for large tail control surfaces. Generally, the aircraft is stable in roll (the dipping wing generates more lift and the aircraft corrects itself).
Another thing to note is that the maneuverability of the ground effect aircraft is pretty limited if it's unable to fly in OGE (Out of Ground Effect). Soviet GE aircraft could fly in OGE for clearing obstacles.
References: Wing in Ground Effect Craft Review by Michael Halloran and Sean O'Meara
