Actually, I'm glad this thread came back, as it's been very entertaining to read everyone's experiences and thoughts.
Mostly I shy from relationships like this at the workplace, school...etc. They really can be awkward when they don't work out or one loses interest, but getting involved with an Aikidoka would seem to be ideal because you have a specific
interest in common. I think the thrill of a dojo relationship would mean having a lifetime aikido partner! You'll get married, build out a permanent dojo mat in a spare room, raise 2.5 little white belts, and ukemi with each other, happily ever after.
Some ideas I had on getting it going without feeling unconfortable...
1) Ask her and a couple of other students if they'd like to practice one day somewhere. That way you are still in a group. When the practice ends, see if she'll stick around to talk when the others have left.
2) Show her your bokken and ask if she wants to try it out sometime.....
3) (I stole this one from another thread) When doing practicing koshi nage (hip throws), tell her she's not "getting you up enough." If she giggles, you have a shot!