As myers said, use a MP3 or cheap cd player, set it to loop so the music plays all the time. and then connect it to a set of amplified computer speaker. Then you can just turn the speakers on or off at the same time as your prop does. The music will start or be heard only when the prop begins to sway. Of couse you will have to have some way to turn the props on and off like a motion sensor, pressure mat or even manually with a switch.
The Cow board are a great product and work really well if you have a sound track that needs to start at the begining each time.
The Pico Boo's have the sound built in and will activate your prop all at the same time. A very good product for a reasonalbe price. Both products cow and pico boo require a trip mechanism. Also both have an ambient sound that plays as people approach then when tripped the selected sound will play.
Take a look at the product review section for feed back on both the Pico Boo's and Cow boards.