This is the second time that I've flown on US Airways.

[Tofu]

Hello Alan, Mister Micawber, Beeesneees, Mordant, Esl_Expert and other native English speakers,

This is the second time that I've flown on US Airways.

Is this sentence grammatically correct?
Is it okay without on?

[Beeesneees]

I think that this structure when talking about a specific airline would be used without 'on' most commonly.
There's nothing incorrect about it as it stands though.
"I've flown by..." might also be heard as a third alternative.

Please don't offer me any airline food as a reward!
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Cheers m' dears!

[Esl_Expert]

In that particular sentence, I think the word 'on' would tend to be included (but the word 'that' would tend to be omitted), in everyday English in my neck of the woods.

The word 'on' tends to be omitted in more formal written English. Also, in advertising sentences such as "Fly US Airways", for example, the word 'on' would not be used.

(I'll pass on the airline food as well.)
:-)
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“The best things in life are silly.” ~ Scott Adams

[Tofu]

Thank you, B and Amy.

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http://www.oxfordadvancedlearnersdictionary.com/dictionary/fly
fly
to travel in an aircraft or a spacecraft
- Is this the first time that you've flown?
- fly (from…) (to…) I'm flying to Hong Kong tomorrow.
- + noun I always fly business class. We're flying KLM.
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Is there anybody who's ever used KLM?