When I fly with my backpack (we backpack hike a lot on vacation), when I get my boarding passes I ask for the giant plastic bag if they have them and TSA will put it in there after it's been checked -- kind of like what they do with child carseats that go through as baggage -- they have done it several times and the straps are fine. The alternative is to tie the straps to each other. I've done that a few times and it works fine.
2013 Specialized Myka FSR Comp
2013 Specialized Ruby Sport (carbon)
2014 Salsa Vaya 3 (steel)
2014 Felt Z75