Pets, particularly dogs (though cats aren't exactly innocent), often like to run around, and that sometimes leads them into the street. If your motorcycle is headed down that street at the same time, though, you can end up in an accident that leaves the pet safe but you in a mess. You would think the pet owner would be liable for your injuries, but not necessarily.
Courts are going to look at whether the animal was properly restrained. If the dog was on a leash, for example, that unexpectedly broke, then the pet owner might not be liable because he or she had the pet restrained in a reasonable manner. A cat owner whose cat dashes out of the house when the owner trips upon walking inside certainly wasn't trying to let the cat out.
However, if that dog leash was already known to be in bad shape -- say, you find neighbors who had repeatedly told the owner to get a new leash because the old one was frayed -- that's a different story. Why hadn't the pet owner gotten a new leash? As for the cat, what if the owner had left the door open out of carelessness despite the cat's continued attempts to run out of the house? In those cases, you might have a valid claim.
The pet's history also plays into the situation. Dogs that have no history of running out of the house or away from the pet owner might not be restrained while in the owner's yard because the owner reasonably expected the dog to stay put. There actually was a case in 1995 (albeit not in Illinois, but it can still be used as a precedent) where a dog owner was found not liable for severe injuries in a motorcycle accident because the dog had no apparent history of running into the street.
What you were doing also plays a role. Were you riding at a slow speed or going the speed limit, paying attention to your surroundings? Or were you showing off, swerving the motorcycle all over the street? Were you speeding so fast that you couldn't stop in a situation where you would be reasonably able to stop had you been going the speed limit?
Courts have to balance all of this information in their decisions. If you have been in a motorcycle accident caused by hitting or swerving to avoid hitting a pet, you need to speak to a lawyer. Your case has to be put together carefully, and a lawyer who specializes in motorcycle accidents is your best hope.