As everyone has said it is hard with a dog but it can be done. We brought our big dog from the UK with us. If I had the benefit of hindsight I would have shipped him a few weeks later and given ourselves more time to look for a decent rental. Decent rentals are hard to find dog or no dog. We also have small kids so needed a place suitable for them which in our case proved the bigger issue. We found finding a tame agent and asking if they knew which landlords would take a house trained dog worked OK.
As it happened we left ourself no time to look for a good rental as the dog arrived 3 days after us so we stuck with a place we found on a bach rental site from the UK. These sites tend to list which ones take dogs but the rent isn't cheap and not financially viable for long term unless you can negotiate a deal.
Debbie