Panting heavily in dogs can be a sign of many things, including pain, fever, nervousness, being over weight, being over heated, or medical problems such as heart or lung disease, Cushing's disease (increased adrenal gland functioning), hypothyroidism (decreased thyroid gland functio) , or narrowed airway.
If the panting is very loud, then I would be checking out the airways first. Older dogs are prone to a condition called laryngeal paralysis where the vocal folds in the back of the throat do not move like they are supposed to, so the air has to go through a narrowed passage way. These dogs will pant very loudly with minimal exertion. Breathing while at rest may be pretty normal. In order to diagnose this condition, the dog has to be sedated and those vocal folds are examined to see if they are moving properly as the dog breaths in and out.
If the breathing is not particularly loud, then blood work checking for Cushing's disease are hypothyroidism would be good to do. You mentioned that he/she tested negative for diabetes , but a general chemistry profile probably would not include adrenal gland and thyroid gland function tests, so these could be done. Most dogs with Cushing's diease will have increased water consumption and an elevation of one of the liver enzymes, but this is not true for all of them. Dogs that are hypothyroid have less tolereance for temperature extremes, so they tend to pant easily.
Older pets often have arthritis-type changes in their joints and this can be a source of discomfort that can result in panting. If you think this may be the case, ask your vet about doing a trial course of some anti-inflammatory type of medications to see if it helps.
I am assuming your vet ruled out fever by taking his temperature and would have discussed weight if this was an issue that could account for the panting. I am also assuming your dog is negative for heartworm and on heartworm preventive. Heartworm disease could also cause excessive panting.
Chest x-rays can be difficult to interpret sometimes, and that is especially true for older dogs. You could request that the chest x-rays be evaluated by a veterinary radiologist to make sure that nothing is being missed on the films.
I hope this gives you some ideas that could be causing your dog's panting. If so, please click Accept. If you need more information, please let me know.