I believe the weight will be necessary because the way the sensor works is to flex the piezo element. So fix the base of the sensor to the dryer, so that the weight can bend the sensor perpendicular to the axis of most vibration. I would guess like this:
If you don’t want any dangly bits, I would suggest an accelerometer… You can even calibrate it looking for the average of the peak acceleration in all axis when the dryer is running (and use the axis that generates the highest value), and then divide that value in half for a good threshold to know when it’s running or not. This can be completely automated in a simple for() loop for calibration.
Some interesting PDFs for your Sparkfun sensor:
http://www.meas-spec.com/downloads/LDT1-028K_Application_Note.pdf (this one talks briefly about a shielded form of the sensor that might be necessary)
EDIT: I checked out my side load washer when it was on the spin cycle… same way a typical dryer works. The vibration seemed worst right in the center of the top panel, due to resonance and the panel being thin. Nearer to the edges of the machine, there was very little vibration. Then again this is a newer machine and everything is probably pretty tight… it usually only “gets crazy” with offset loads. Still though, with a cantilever style vibration sensor like the Sparkfun one… you may need to adjust it for your specify vibration frequency for best sensitivity. I still think the accelerometer is great universal way to go.