I originally said this effect wasn't translatable to GIMP, but after further poking about I figured out how to do a variation on it. Tutorial made using GIMP 2.2; for advanced-level/confident users.
(PSP and Photoshop Elements versions archived here.)
...The bad news is that although GIMP does have a function for entering custom filters, you can't save them; so any time you want to use this you're going to have to type the numbers in again. (The current settings will stay in memory while you've got GIMP open, but they'll be wiped away if you close the program and reopen it.) Sorry, folks.
0: Choosing a Base
This effect works well on some images and not so well on others. As a general rule, you want to go for contrasting colours, well framed faces and fairly flat lighting. (Dark-haired people with dark clothes against a coloured backdrop? Good. Pale-faced blondes in winter landscapes? Not so good.) Blurry images, busy backgrounds and strong patterns of light and shade can often mess the effect up. But then again, sometimes they look cool. It really is mostly a case of try it and see.
Once you've picked out your base image, crop it to a square but don't resize it yet - you want it as big as possible. (The image shown below is half-size; the original I'm working with is about 500 by 500 pixels. The full size version is here, for those who want to play along at home.)
1: Creating the Custom Filter
From the menu, choose Filters > Generic > Convolution Matrix. You'll get a dialog with a five by five grid of numbers.
Fill it out with the following set of numbers:
Set the Divisor to 1 and the Offset to 0. Choose 'Extend' as the border type, and make sure the Red, Green and Blue channels are all checked and none of the other boxes are. Click OK to apply the filter.
That's quite an interesting look just on its own, but it's very messy at icon size, so I prefer to filter out some of the noise before making it any smaller.
2: Reducing the Noise
Select Filters > Enhance > NL Filter. Set the Filter type on 'Alpha trimmed mean', and drag both the Alpha and Radius sliders over to 1.00. Click OK.
2b.That's smoothed it a little, but it's still quite speckly, so repeat the NL Filter again with the same settings.
3: Finishing Up
Now you can shrink the image down to icon size.
That's basically it, but the colours end up a little washed out, so I usually duplicate the base layer and set the mode to multiply.
And that's it! When you save the file, remember to do it as a GIF - this effect looks fine in GIF, and it'll be a much smaller file than if you use PNG or JPG format.