Most places around here have outside seating, and dogs are usually welcome to sit outside.
However, my idea of an unobtrusive dog doesn't seem to coincide with what others consider unobtrusive.
When we were in a very dog friendly town last weekend, we sat at a table outside and a man with 4 dogs with questionable leash manners sat at a table a good bit away from us, his dogs periodically barking at other dogs, passers by, the air... It wasn't constant, but enough that it wasn't exactly pleasant either.
I've also seen people let their dogs sit in their laps and put their front feet on the table or lick plates. I mean, I know the restaurant is going to clean the plates in the deep cleaner steam thing all restaurants use, but still, that's not okay IMO.
Penny did sit on my lap but no way was she touching my food, and no way would I let her put her feet on the table either.
I get wanting to take your dog with you, but not all dogs are ready yet to go out in public like that.