This method is most effective on low detail images but can work on anything really depending on the level of removal that you need (of course it can't work miracles). You can also use this method for removing general noise from an image, not just MPEG or JPEG compression artifacts.

Firstly we're going to use the GNU Image Manipulation Program (GIMP) to edit the image, so if you haven't already got it installed you can download it for free from their website. GIMP is available for Windows, MacOS and Linux so that should be pretty much everyone covered.

Once you have GIMP installed, open up the image you'd like to clear up. The tool we'll be using is the Selective Gaussian Blur filter available under Filters >> Blur. So now select either the area you would like to clear up (or don't select anything if you want to work on the whole image) and select the Selective Gaussian Blur tool from the Filter menu.

You should now be presented with a window like the one below:



You can now tweak the Max Delta value to apply different levels of 'cleaning' to the image. Higher numbers remove more details while lower numbers keep smaller details at the cost of not removing as much artifacting.

Below is a close-up of an image with a Delta of 9 applied. The original is on the left:



As you can see, it's eliminated a good deal of the artifacting but keeps the sharp edges of the image in tact. This is very useful on photographs taken with a high ISO (which generally causes a large amount of random noise, especially on cheaper cameras) or on comic/line art style images such as our example from South Park.

It may not be that easy to notice at first as the artifacting wasn't that substantial to begin with, but it serves as a good base to show what this tool can do. It's also useful if you want to save an image to PNG format. Why? Because removing the artifacts and random out of place pixels means that the PNG compression will work much better and you will end up with considerably smaller images. A win all round.

Below are the two full images, before and after the effect has been added:




So there you have it, a simple tool to remove artifacting, grain and noise from images using a free image editing application.