Generally thats how its done (left right and such). You'll notice a lot of restaurants check have a diagram on the kitchen copy, that shows how you are supposed to serve a table, and where the server stands, but I think that not only servers, but society has slacked on that in a big way.
Some tables dont care what way you go or where you start to order. A lot of people order 'ladies first', so no matter where they sit, all the women order, and then the men. Some want to be cleared immediately after finishing, some want to wait until everyones done.
If you are working in an establishment that feels they are that upscale to go by the old school rules, they should make that clear from get go. Otherwise, as they say, the customer is always right, and you order and clear however is easiest for them.