I think the definitive article on the subject is "Cleaning your Sensor""Cleaning your Sensor" by Thom Hogan, written by someone reputable with a lot of experience. I would disregard most scary personal anecdotes and product reviews by people who have given this one try, quite possibly misused the product, and then ranted about it.
It is safe to clean your sensor yourself with the right products with care. If you are not careful, use the wrong products, or don't follow instructions, it's easy to end up just moving the dust around or picking up oil from surrounding areas and spreading it across the surface and making things worse (and then claiming that the products don't work!)
There are three main contaminants that you have to deal with:
light dust clinging to the sensor due to static charge - this is often taken care of by the camera's cleaning function, or can be blown off by a rocket blower
heavier dust "stuck" on the sensor, which can usually be removed with a dry brush
oil, and dust that is really, really stuck on - these usually require wet wipes to remove
Your camera's built-in sensor cleaning function, along with a rocket blower, will remove most light dust. Most cameras can be set so the cleaning function is run whenever the camera is powered on or off.
For heavier dust you have two options: - dry brush, and wet wipes/swabs
dry brush
this method uses a statically charged, dry brush (such as the Arctic Butterfly) to attract the dust.
A light brushing against the filter surface may be required to dislodge the dust. You must be very careful not to touch the bristles of the brush against any surface other than the IR filter itself, or you may pick up oil or other contaminants and spread them onto your sensor.
The advantage of a dry brush is that it is reusable and effective on most light dust. If properly used it will not leave any residue on the surface.
After using the dry brush, use the camera's in-built cleaning function a few times to pick up any dust that was loosened but not removed off the surface.
wet wipes/swabs
For oil, or really stubborn dust, the final course of action is to use wet wipes, such as "Sensor Swabs".
Again, you must take care not to touch these against any surface other than the IR filter to avoid picking up dirt, dust or oil. You need to buy a swab that is the right size for your sensor, and swipe once or twice across the surface, then do not reuse. These swabs require a solution - make sure to use whatever solution is recommended (e.g. Eclipse) and do not use too much solution to avoid leaving residue on the filter surface.
These swabs tend to be pricey - if you are good and efficient at cleaning, they are worth it -otherwise you may use up 4-6 of these getting your sensor free of dust. So depending on your confidence and steady hands, you may want to consider having the sensor professionally cleaned if a rocket blower or dry brush cannot dislodge the dust.
There are also sensor "pen" products. I have not used them. Be careful not to use a "lens pen", PEC*PADs, or any other device that is made to clean lenses or LCDs. Make sure the product is designed specifically for sensors.