An Neutral Density (ND) graduated filter is probably the easiest way to inject a bit of additional drama into a crappy, lifeless sky. Also, helps you bring the exposure down, which can be a problem when the sky is many stops brighter than the foreground.
Shooting in mono (or converting in software afterwards) is also a good way to look at things differently, especially for creating high contrast and silhouettes. That's another option...
Personally, I use ND filters or where I can (on my 14mm for example which doesn't accept them) I add them in in LR3 to at least give the sky a bit of punch. It's a bit of a folly to rule out processing because it can help so much.
I actually like the muted tones you get from overcast skies and more often than not, the light is fantastic for outside portraiture and details shots because it's so diffused. Add a bit of off-camera flash or a reflector (especially a gold one) and you can create some stunning stuff. I also find that those conditions lend themselves to long lens landscapery because you you get different densities of haze that can help create some really graphic images.