This is the simplest way that I’ve found to make the background colour of an image transparent.

  1. Download GIMP
  2. Open your image. I’ll pick this one:
  3. Optional: Select Image – Mode – RGB if it’s not RGB.
  4. Select Colors – Colors to Alpha…
  5. Click on the white button next to “From” and select the eye-dropper.
  6. Pick the green colour on the image, and click OK

The anti-aliasing is preserved as well.