One of the sounds commonly associated with "old-time" banjo playing is a rhythmic "clucking" noise on the back beat (beats 2 & 4 of each measure). Learning to produce this sound takes a bit of trial and error, but it is really simple:
Playing with the right hand in the correct place is very important--essentially, we want to be playing over a natural harmonic on the string; the most common place to sound the cluck, therefore is over the 19th fret. This is why many banjos used for old-time music are "scooped," i.e., the frets above the 17th have been removed and the fingerboard lowered to make it easier to play over the 19th fret's former position.
On the "brush" beat, instead of striking the strings downward using a single finger, drop the right wrist so that the the fingernails are almost perpendicular to the strings and directly over the 19th fret.
Instead of striking the string with the usual finger, begin the brush with the next finger: i.e., if you usually play with your middle finger, strike the strings first with your ring finger. (If you play with your index finger, you'd begin with the middle finger).
Follow through immediately with the usual picking finger. In this way, the initial strike sets the string in motion, and the following finger cuts the sound off, leaving a harmonic"cluck."
To further emphasize this sound, the heel of the right hand can be immediately dropped onto the vibrating strings to mute them, producing a chopped sound to the cluck.
Watch this video clip showing this motion.
While the ClearHead™ video shows these motions (exaggerated), it cannot reflect the correct sound, as playing over the transparent head makes my hand miss the harmonic spots! Thus, look at the first part of the ClearHead™ video clip to see the technique, but listen to the sound achieved in the 2nd part of the video, when my hand is up over the neck!