My first digital SLR was the D70, and dirt on the sensor was a problem. I bought some Eclipse solution and cleaning paddles, and held my breath. Found it really wasn't difficult at all. Used the same supplies on my later D200.
When I got my D800, I had to get larger cleaning paddles for the FX sensor. I bought the items that Thom Hogan recommended. My kit now has paddles, two different liquid cleaners (one for oil and one for other contaminants), a brush and an anti-static bulb blower.
I had some stubborn gunk on my sensor after some shooting, and wet-cleaned it using the regular solvent. Since then, I've only needed to do periodic blow-outs using the bulb. The brush hasn't been needed yet. The bulb also does a fine job on the focus screen and mirror.
I know it's scary to be diddling with the most expensive part of the camera, but using the right tools in a relatively clean environment shouldn't result in any problems. Be careful, be gentle, and be patient.