The manual recommends using a blower when the automatic cleaning fails - or better yet, sending it to Canon for service. Sometimes - rarely - a blower will work and is the simplest and reasonably safe thing to use.
Otherwise many like the Sensor Swab solution. It is tedious and expensive, but I have used it effectively several times - especially on the 5D that seemed more prone to spots than the newer models. The problem I always have is going from bad to worse to finally better - a lot of patience is required and wasting or going through several of the swabs is not uncommon. Maybe my technique sucks, but I find it takes a lot of time and swabs to be eventually be satisfied. That said, I have never caused any damage and I have always managed to finally get the sensor clean. Along with the swabs, good lighting and magnification (reading glasses or similar) is also necessary as the spots that look huge on a photo will be nearly invisible.
http://www.sensorswab.com/
Steve
EOS 5D MKIII
