Like Scott & Sparky, I use
(a) the blower
(b) the inspection loupe
(c) sensor swabs and Eclipse solution
The sensor is normally always covered except for those brief moments when you click the shutter. Sparky is right... there are filters behind the shutter but in front of the sensor. You can't actually touch the sensor without a disassembling quite a bit of the camera, but you can readily access the filter in front of the sensor.
But as the sensor is typically only open for a brief moment in time, what tends to get on it is the occasionally particle of loose dust. A couple of cameras have had oil splatter (this is mostly limited to a few isolated models... the D600, for example, had the issue, Nikon corrected it when they came out with the D610).
With that in mind, usually the camera's self-cleaning cycle will shake the dust off the sensor, but I point the camera face down when so that the dust doesn't just drift down to a lower spot still on the sensor (dust is light... it'll drift in the air and that's how it found it's way onto the "sensor" in the first place.) Often times that's all it takes.
If that doesn't work, I use the hand-squeezed blower (I avoid compressed air because if it isn't perfectly level, the propellent can spew out and it leaves a residue. The residue looks like a fog that won't evaporate and then it does have to be wet-cleaned. )
If self-cleaning and a hand-squeezed blower can't clear whatever is on the sensor, I become increasingly curious as to find out what could possibly made it onto the sensor which is so stubborn. At this point I use the magnifying loupe to inspect the sensor surface just so I can see what I'm working with... then I break out the sensor swab with a few drops of Eclipse solution to clean it, then re-inspect with the loupe to make sure it's clean. Eclipse is nearly pure methanol. It'll break up whatever was on the surface, but the solution itself evaporates VERY quickly and leaves no residue behind (you need not fear streaks, etc.)
To test for sensor dust, use a very high focal ratio (e.g. set your camera to aperture priority and dial in f/22) and take a photo of a plain non-contrast surface (a white wall or ceiling... a cloudless blue sky, etc.) Don't worry about shutter speed... it doesn't matter if you have camera movement due to long shutter exposures (it probably actually helps). This causes the shadow of any dust on the sensor to be well-defined and visible in the image. The higher the f-stop, the easier it is to find the dust spots.