Piezo or vibration sensors

You have to decide what exactly you are trying to detect. You say "horizontal rocking" which implies a pivoting motion, but then "horizontal plane movements" which is something else. You need to make up your mind. I would detect these two differently.

Also, what exactly is "movement". Do you need to know the distance your object has moved, it's speed, or will acceleration do?

Acceleration is the easiest of these to measure with small and cheap off the shelf sensors. A number of companies, including Freescale and Analog Devices, make MEMS accelerometers. These come in flavors that either put out a analog signal proportional to acceleration, or also digital. You can use two single-axis accelerometers, or it may be more convenient to use one two-axis chip to detect motion in a plane. With the two axes aligned along the plane, the total acceleration magnitude is the square root of the sum of the squares of the individual acceleration signals. If you're just looking for some threshold, then you can square the individual signals, add them, and compare that to the square of the acceleration threshold.

You can do limited itegration of such acceleration signals. There is enough offset and hysteresis in cheap MEMS accelerometers that inertial navigation for more than a second or two becomes so inaccurate as to be unusable for most purposes.

There are accelerometers that are designed with the application in mind. There are many options available out there but a couple that might be useful:

• Raise an output high whenever acceleration is above a certain amount and lower when acceleration stops.
• Raise output high when acceleration is above a certain amount and output is kept high until a command is sent to clear.

Both these are designed for this application to give you as simple a circuit as possible and I suggest them compared to other options but you might have reasons to need a mechanic tilt switch.