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I have flown on AirTran airways on three round trips!
I flew from New York to Akron, Ohio (and back).
From Boston to Baltimore (and back).
From Boston to Rochester (and back).
They are certainly as safe as any other airline. Depending on the route you are taking, you'll either be flying in a Boeing-717 or Boeing-737, both of which are extremely safe planes. In addition, they are newer than the average plane flying in the world today.
As far as flying in general, it is one of the safest modes of transportation. Personally, I have flown 252 times, and never once did the plane have any problems. One of my friends has had some issues with flying, and he saw his doctor, who prescribed some mild anxiety pills for each time he flew.
Here are some articles and sites: http://www.nytimes.com/2007/07/24/health/psychology/24fear.html, http://www.guidetopsychology.com/fearfly.htm#9 , http://www.airtran.com/aircraft/new_717_jets.aspx , http://www.airtran.com/aircraft/737_background.aspx
Let me know if that helps you out. If so, thanks for pressing "Accept". Feel free to reply with any concerns.