I think Naomi is ok but sounds a dated. I know quite a few who are around my age (late 30s).
Poppy is nice I don't agree with people saying it's childish I know full grown women called Poppy, Daisy and Rosie and I don't think of any of them as childish.
I love Romy, it's my favourite and makes me think of Romy Schneider. It's also the most unusual of the three, so not everyone will know it and how to pronounce it whereas they'll be familiar with Poppy and Naomi so it depends how you feel about that. Having said that the Naomi's I know pronounce it differently and that seems to cause some confusion too.