In the meantime, here are two workarounds for Opera:
1. when you are doing a flight search (or hotel search), wait for the little green bar to progress all the way across. Then hit refresh. Your flights will appear!
2. Try basic mode kayak. It's kind of, um, basic, but it works with almost all browsers. Even really old ones. No AJAXy goodness, of course. Mr. Miyagi says: basic on, basic off.