Thank you for your response.
This section does indeed point out that a dog is considered personal property and creates liability on the owner for a dog bite. However, it does not create strict liability for the dog owner in a situation like yours.
The ancient common law that you are referring to is that the owner of livestock is strictly liable for any damage done by the livestock trespassing on another's property. This ancient common law has been subsummed by the Nebraska statutory law:
§ 54-401. Estrays, trespassing animals; damages; liability.
The owners of cattle, horses, mules, swine, sheep, and
goats in this state are liable for all damages done by such stock upon the lands
of another in this state as provided by section
54-402 if the damages to the lands are not the result of negligent or
willful damage to the division fence by the person claiming damages to the land.
This law does not apply to dogs that are off the leash and running on public property.
I'm afraid that your situation may fall under the doctrine of Unavoidable Accident, in the the dog will be treated as a hazard on the road and it will likely be very hard for you to recover from the dog owner.
However, you would have to proceed on general Negligence principles. In other words, you would sue the dog owner and try to prove that the owner had a duty to keep the dog on a leash or behind a fence. You would point to Section 54-608 which requires that dogs cannot be at large in counties with 80,000 inhabitants or more. You would argue that this creates a duty that the dog cannot be left to run at large and that the owner is responsible for any damages that resulted.
Again, I think you have a very hard case, but this is the argument I would recommend if you are going to try and make the claim.