Mixed breed dogs can now compete in some AKC events like Agility, Rally and obedience if the dog is enrolled in the AKC Canine Partner Program. However, you would have to specifically search for events allowing mixed breed dogs to compete. From my understanding, the different clubs sponsoring each AKC event are allowed to decide whether or not they will open the event to Canine Partner Program dogs.
Also, if I remember correctly, inorder to enroll your dog it must be spayed/neutered.
This is the link to the AKC Canine Partner Program.
AKC Canine Partners. AKC program for Canine Partners.
You can also sign your dog up so that you can compete in NADAC (North American Dog Agility Council) events. All dogs (mixed or purebred) are able to compete.
Here is their site.
North American Dog Agility Council, Inc. (NADAC)