@beetgrob: I honestly don't know why we say "do the washing" as well...but you're right, it's expressing the same idea. There is no strict rule, it is up to you. If you want to be specific you would say "do the laundry" or "do the dishes".
I believe that "do the laundry" is more commonly used in American English, and "do the washing" is more commonly used in British English. I haven't heard "do the washing" used in reference to dishes, only clothes/towels etc., however if you say you are "doing the washing up", that means dishes. The "up" makes a big difference! :)
"Mum, have you done the washing? I have nothing to wear!" "Mom, have you done the laundry? I have nothing to wear!"
"Once you've finished your dinner, I'd like you to do the washing up please." "Once you've finished your dinner, I'd like you to do/wash the dishes please."
@beetgrob: yes, pretty much! No, but you might hear someone say "I'm just going to wash up" or (more likely) "I'm just going to freshen up". There's no use of the verb 'to do' in this context. Again, that's American English.