We do it monthly, which I would recommend instead of weekly. We’ve been doing it a year and a half now and it’s went over really well. It gives the regulars something new to try, and it also gives people something to talk to their friends about.
We’ve added some things to the menu because they went over so well, and we’ve also made some items for a particular season and now we are ‘bringing them back by popular demand’ a year later. I personally think weekly would be a lot of work, but coming up with something new every month is feasable.