Since it is on each picture and independent of lenses, it sounds like dust/other on the sensor. It tends to be easily seen at apertures of f/11 and smaller. However, an example picture would be helpful in the diagnosis.
Assuming it is dust, you will have three levels of cleaning to try if you want to do it yourself: (1) Use a blower like the Giottos Rocket blower. (2) Do a dry cleaning with a brush. (3) Do a wet cleaning.
Yep, that's definitely a drop from the lubrication of the shutter. Wet clean with anti oil liquid will be the only answer. I had the same thing with my D300s and had to bite the bullet and clean. It's scary at first, but as long as you are careful and not too trembly handed you should be fine.