What would probably be ideal for this kind of application would be a linear motor with some kind of positional feedback (optical, most likely):http://en.wikipedia.org/wiki/Linear_motor
Unfortunately, I've never found a source for linear motors where they didn't cost a fortune for any appreciable stroke length.
You might be able to do something with a DC motor connected fairly directly to a rack and pinion system; basically a form of a linear actuator, but without the gearing down that is common for those devices. I would think it would have to be fairly precision designed to keep it mostly quiet and long running (likely using well lubricated nylon or other plastic gearing).
Another possibility (hinted at by Grumpy_Mike) would be a to use a scotch yoke or other crank mechanism; the only problem there is that at the top and bottom of the stroke, there is non-linearities involved; I'm not sure if in your application it will matter, though.
All of these solutions may require some kind of positional feedback, once again depending on what you are trying to do (if you are just trying to randomly flutter the cloth, no positional feedback is likely required - but if you are wanting to try to set of standing waves or such between the "nodes", knowing which actuator is where would probably be important).