First, we fill the canvas using the hurl filter (Filters > Noise > Hurl). Set Randomization (%) to 100.
This will do the same thing as the Scatter RGB filter (Filters > Noise > Scatter RGB) with all channels (Red, Green, and Blue) set to 1.0, However, Scatter RGB has slightly more options, and if you turn off Independent RGB in the settings dialogue, may allow you to skip the desaturation step.
Then, apply a motion blur to the image (Filters > Blur > Motion Blur...) with an angle of 0 and a length of about 50. Change the angle to 180 and repeat the effect. Why? Notice how the right edge of the final image isn't nice and brushed looking? Doing it in both directions will smooth this out. You may have to adjust the length setting so as not to "overblur".
Alternatively, you can use the Gaussian Blur filter (Filters > Blur > Gaussian Blur). Just click the chain icon and turn the vertical value down and adjust the horizontal value until you are satisfied. This effect will not leave a side of the image left unblurred and may have an even more satisfying final result with proper adjustment.
This is the shortest step. Now we only need to desaturate the image. Go to Colours > Desaturate and press Lightness, although Luminosity may sometimes look better.
Apply cropping / tiling / brightness / lighting filters to suit. Cropping may not be necessary if you've used the Gaussian Blur approach as opposed to the Motion Blur approach.