The problem with a dog travelling along with a bike is that the dog will likely be able to go much faster on the climbs and the bike much faster on the descents. I found that for short climbs and mountain biking it was manageable for me and my dogs. I left her home when I toured. Trail running worked out better than biking for us and my current dog and I probably ran close to 20,000 trail miles together (I lost count at 10k miles but we ran for a long time after that). She has also been a great hiking/backpacking companion too.
In my case I am pretty sure my pooch was happier being home with my wife than she would have been cranking out long miles on the road for months at a time, so she stayed home when I toured.
On the road and in country with very long climbs and descents it gets logistically more difficult to deal with a dog. Not impossible just more difficult. I met a guy who was travelling full time with a greyhound saluki mix. He said the pooch rode in the trailer most of the day and got kicked out for the climbs to run along side. He said that worked well for them.