Instead of leaving the camera to full automatic, I set it to aperture priority, meaning that I only need to set the aperture and the camera will select the shutter speed for me. Most advanced cameras have this feature, but on the Fujifilm X-E1 the lenses have their own aperture rings which makes selection very intuitive. Changing the aperture means that you can control the look of your image, whether you want to isolate your subjects with a wide aperture or ensure maximum sharpness across the depth of the image with a small aperture. Most of the time I leave my aperture at the wider settings because I like the "filmic" look of shallow depth of field images.
The problem with letting your camera select your shutter speed is that it can select shutter speeds that are too slow to handhold. Sometimes at wide apertures, I would not have a shutter speed fast enough to balance all the light coming in.