Occasionally PRA (progressive retinal atrophy); gastric torsion and bloat; canine hypothyroidism; and hip dysplasia. Bloat is a health issue to most dogs, being the second largest killer of dogs other than cancer, but Briards can be particularly susceptible to it because of their deep chests.
There are several good articles on bloat, hypothyroidism and hip dysplasia available from the Atlantic States Briard Club .
Ironically, although Bones and Houlette passed over the rainbow bridge before their time, they did not contract any of the health issues assoicated with the breed, so I cannot offer any first hand information. Bones died of lymphoma and Houlette died of complications arising from a failed laryngeal paralysis surgery.