What you have in the images there is called a
Moiré pattern.
Simplest way to minimise it is to scan from the source at 50%, or scan at default 100% and reduce 50% in an app like Photoshop or GIMP. If all you're doing is putting that scan up on the 'net, you could also try scanning at 144dpi and reducing to 72, but that doesn't always yield good results, depending on the source material. A small-scale (.5 to 1px) Gaussian Blur may also help. It won't completely remove the pattern, and you might lose some ultra-fine detail (which would likely not be there anyway in a previously-printed image), but it's usually good enough for 'net use. Watch copyrights!