I cannot echo what Dr Greg has said enough. When I arrived in November '09 I stupidly arrived on an L-visa under the false pretense that it would be converted when I had arrived, which never materialised. Then you have to do regular visa runs to Hong Kong to get a new L-visa just so you don't overstay, and you are still working illegally. You regularly hear schools promising they will sort it for you, and they never do--they will sometimes tell you that you have to pass a probationary period first.
I am in this exact Catch-22 at present as many schools will not hire me without a working visa, but are also unable/ unwilling to get me one themselves meaning I have to do most of my work freelance and tutoring privately. This is not a bad arrangement financially, but is still illegal.
You can teach legally in China without a degree but I will tell you one thing: it is difficult to do this in Guangdong province as the government's minimum requirements tend to be enforced, especially when the government decides to have a sporadic crackdown, like during the recent Asian games.
P.S. Dr Greg, could you please look at my recent topic "getting a work visa on a tourist visa" on Ken's page and give me your input, it would be much appreciated.