What is the best way to introduce the Six Sigma approach to non-believers?

One sentence. Don't use the word Six Sigma.

Six Sigma has gotten a big rap as another big thing that is going to fall off the planet just like most things have. In reality, as I see other methodologies evolve, they all gravitate to the same principle -- of doing projects for the customer. Work the business case so it demonstrates, very clearly, the value that will be derived by your customer. Very important, take the time to clearly state this business case. I see this professed again in the new ITIL V3 Service Strategy phase, for one.

As Six Sigma professionals, we should not be so arrogant as to assume that doing business this way is only our domain. It is a reality that other good managers have done it this way, even before the terminology was coined so.