Shoot from inside a car (window down, of course). Shoot from under the eaves of buildings. Shoot interiors - the soft diffused light of an overcast day is magic in some buildings - you'll need a tripod, or if tripods aren't allowed its a wonderful opportunity to see how else you can steady your camera - take it as a challenge, eh? Stick an umbrella (a big one) on a tripod, and then stand under the umbrella - though it doesn't work on windy days. Set-up a table top studio at home and shoot still-lifes.
I feel your pain... I'm in the middle of temperate rainforest.