I could definitely see a wider variety of pigeons and doves being introduced, though before the late 1800's North America had its fair share of the pigeon-niche with the Passenger Pigeon...
there has been a smaller introduction that has been very successful - the house sparrow
http://en.wikipedia.org/wiki/House_Sparrow
budgies could do well in several places
http://en.wikipedia.org/wiki/Budgerigar