You can create a content processor that will do this for you. Here is an example of a simple texture processor. This is for premultiplied alpha (which XNA 4.0 already does, but it was useful in 3.1).
You could easily modify it to read in the first pixel and clear all the pixels with matching values. Something like this:
Color m = bitmap.GetPixel(0, 0);
for (int y = 0; y < bitmap.Height; y++)
{
for (int x = 0; x < bitmap.Width; x++)
{
if (m == bitmap.GetPixel(x, y))
bitmap.SetPixel(x, y, Color.Transparent);
}
}
Put this in a new Windows Library project, add it as a reference to your content project, build it once, and you should be able to select it from the processor options on the properties pane (F4) for your textures.
Of course, if all your sprites have the same background colour, jv42's answer is easier.