I am a passive aggressive note-writer myself.
Since it's soon enough after you moved in that you don't have chore-doing logistics figured out yet, you could say "Do you think we should each be responsible for washing our own dishes, or should we take turns doing them?" Maybe a little passive aggressive, but I prefer to think of it as tactful :)
I like Julie's idea.
i honestly could not say, I do all the dishes at my apartment, but also I don't mind.
Could you try aggressive notes?
You could try asking them ;)