This has been a huge issue since I can remember. I run into it regularly, and figured I should drop some knowledge on the internets, for those who also have this issue and can't figure this out.
Do you see in the before the nasty banding that occurs? This can occur in a variety of situations. I at first thought it had to do with my color space, or sample rate, but it occurs in all 8, 16 and 32 bit spaces, CMYK and RGB. So no way to really avoid it, but you can clean it up. If you are seeing banding in your images, go to your Filter Gallery and find the Spatter filter in the brushstrokes setting. Spray radius turn all the way up to 25, and the smoothness to 1.
A few things to remember. The first, in order for this to work, the layer must not have active layer styles applied. If you do, simply create a new layer underneath and merge down (cmd-e, or ctrl-e on windows) Then apply this filter.
The second thing to remember is this will add a little bit of texture. I find the texture you get from this effect is a fine compromise, but it may not be for you, or your particular situation.
Last, adding this effect will make your file larger. This has to do with how the compression occurs in various formats. I won't go over the specifics on how that all works, but do take note of this. It may not matter. It might matter a lot if you are working with web graphics. Keep that in mind, and as always, this is your design. Make it your own. If you find any improvements on this technique, or better way to do it, leave a comment!