My first encounter with a dog chasing after my bike was on my first poker run.
A big german shepard ran out into the pack of bikes I was riding with and tried 
to bite the right leg of the guy riding in front of me. He kicked the dog a 
couple of times (we were all only going about 10-15mph) and just made the dog
angrier and got away by applying WFO and outrunning the dog who then started
after yours truly. At the time, I had open drag pipes (obnoxiously loud as hell)
and as soon as the dog was about 2 feet away from the right side of my bike, I
pulled in the clutch and revved the motor up to about 5000rpm and the roar of
the pipes frightened the dog away (probably made the dog's ears hurt like hell)
and he didn't come back for the rest of the bikers behind me either. 

My second encounter was with an already deceased road-kill dog. I wasn't paying 
very close attention to the pavement when the car in front of me straddled the
carcass. I looked down just in time to see what I was about to run over so I 
just held onto the handlebars and freaked out. Both wheels went squarely over
the dog's belly with a thump-thump just like running over a piece of a 4x4
lumber. The bike didn't lose any stability at all and I kept on going. After
I got to my destination, I found bits of fur, meat, and blood stuck to
the bottom of my motor and frame...... gross.