While trying to change the color of an arrow graphic (that I only had in a flattened format), I was trying to remove the background so I could apply some color fills. I had no luck getting rid of all the jagged edges until I came across this gem from Linda Nicholls:
Apply the Convert to Alpha filter. Filters > Other > Convert to Alpha.
I haven’t done enough playing around to know if this only works if your background color is the same as the color you want to make transparent, but it worked perfectly on my image: