I see this a lot:
I shall say, Daphne, that my going to Athens on Saturday need not disturb our tea schedule one bit.
Versus what seems right to me:
I shall say, Daphne, that me going to Athens on Saturday need not disturb our tea schedule one bit.
I'm somewhat sure that the former is actually the right one, but the latter always seems like the right one to me.
Can both be used? If seems weird to say "my going to Athens" as opposed to "me going to Athens", but the more I use "me" in this context, the less I sound like a native English speaker (which I'm not) and it bothers me.