A recent camera club project asked the members to photograph a subject from as many angles as possible and then select five different viewpoints to show at the next meeting. My favorites are included in this post, but the main reason I'm sharing them is that I returned to a technique I've only dabbled with in the past --- purposely not looking through the viewfinder or at the LCD screen when shooting.
The main reason was necessity. I was intent on photographing some daisies in a public flower bed and didn't want to trample on them. But the angle I was most interested in created backlighting that made the petals translucent. I wanted a low angle close to the flowers, but I couldn't lie down in the flower bed to get my result.
Instead I set my compact digital camera on macro focusing and just held the camera below the flowers, pointing the lens in the appropriate direction. I pressed the shutter button halfway, listened for a double-beep that confirmed that the camera was able to focus, then took the shot. I checked the results on the screen, then tried again, adjusting the camera position according to the results I got the shot before. My favorite is the one below.
In this particular instance, I was shooting below my usual eye level, but I could just as easily have held the camera at arm's length above the flowers (an angle I didn't try) for a different perspective.
If you like to take candid street photos, you can literally "shoot from the hip" by holding your camera at your side and taking the shot. In this situation, you might also want to turn on continuous shooting or burst mode to take a series of photos.
It's the compact digital camera's design that makes this so easy. (You can also do it with a digital SLR and a wide angle lens; it's just heavier.) My camera (a Canon Powershot A570 IS) is lightweight and easy to hold with one hand. All compact digital cameras have lots of depth of field (the amount of the scene from near to far that appears in focus), especially when the zoom is set to wide angle. This ensures that you can get a photo with nearly everything in focus, even if you aren't sure where the camera is focusing.
So the next time you think you can't get the shot because you can't get your body into position, think again. If you can put the camera in the right place, you may get the shot, or an even better one, after all!