Sorry if this is overtly obvious, but having taught for many years, I've run into this with students many times. The problem usually comes when the student (player, in this case actually) isn't touching the string EXACTLY over the correct spot (12th fret for instance). If your finger is just a millimeter or two away from the harmonic node, you won't get a good clean sound out of it. Also, as soon as you pluck the string, it's best to get your left hand finger off the string so that it won't dampen the vibration of the harmonic. JMX's post lists the nodal points correctly, but remember some of them are pretty tricky to get at first. Start with the ones at the 12th, 7th, and 5th first, once you get those clean, you can start messing around with the others.