Qatar is probably the largest producer . Nigeria also has several trains working, the Angolan LNG site has a really cool intro that makes it worth a visit just for that, and Algeria has been doing it the longest . There is a more comprehensive list here

Oops, I also missed Australia , Yemen which all of a sudden is having some problems, and Iran . At one time Indonesia was very prominent, but it has been having problems and has not been fulfilling some contracts, as well as having domestic use issues.

Here's some factors which could undercut Australia's LNG exports when present contracts have expired;
1) a northwest to southeast transcontinental pipeline
2) large truck fleets switching from diesel to CNG
3) fear of future carbon taxation leading to more combined cycle gas fired generation and backup for renewables.