It (almost) never fails. Whenever I'm half way around the world at someplace photogenic, and at a time of year when "it never rains here", I get rain. Or at least overcast skies. There are any number of arguments for and against the benefit of a solid gray sky, but I'm soundly on the "against" side of things. Actually, I'll admit that there are a handful of subjects or/and situations that are easier to shoot under the cloud (notice I said cloud and not clouds - if there are clouds that means there is some blue sky showing through, and that's OK).
The first thing necessary when shooting in overcast conditions is to be very careful about how you meter the scene. If you use average (overall) metering, you will invariably get a dark subject under a perfectly exposed 18% gray swath of sky - not what you are looking for. What to do? Use spot metering, if your camera has that feature, to set the exposure on whatever the subject is. Absent that, adjust the camera to overexpose by at least one EV (f/stop). It won't matter if the sky comes out absolutely white; there's no important detail in it anyway. You need to get the subject properly exposed.
Eliminating as much of the sky as possible can really help things out. By positioning yourself to use trees, shrubs, signs, buildings etc., to occupy the area of the image that would otherwise be sky helps in two ways. First, it makes metering easier since some, most, all of the sky is no longer influencing things. Second, just about anything other than that blank gray stuff is more interesting to look at. This is when you want to harken back to the post about leading lines and light/dark areas.
The one subject that comes to mind that is actually helped by overcast skies is flowers. The overcast eliminates the directional nature of sunlight and provides a flat, even illumination. This gives all the subtle detail of the blossoms an even look without competing with shadows caused by petals and leaves blocking the light. Just about every good photo you see of flowers taken outdoors is either under overcast skies, in the shadow of a building or overhang, or under a portable light diffuser the photographer brought along.