And where else if not McDonald's! Your dog will not receive any special treatment, but at least you can always be sure that there are plenty of tables to sit outside. It can be a little tricky to eat with your hands (since, of course, you have to use one of them for petting your dog) and remember that you can not come to a place alone with your dog since it is a self service and orders need to be placed inside where dogs are not allowed. Other than that, the rest should be easily manageable.
Remember that the place is usually full of kids, who may be scared of dogs, so you should be cautious and act accordingly.
We have taken our dog to only Athalassa and Palloutiorisa McDonald's and haven't tried out others yet. The rest should be no problem either, except for the one in Ledras, which might be too crowdy for a dog (OK, don't count on the one in Mall of Cyprus either 
).