Are there any good pet or dog-friendly golf resorts in the southeast?
The words "golf resort" in your question make it a little tougher to answer. While there are plenty of pet and dog friendly hotels near golf courses, they are not true golf resorts, where the courses are right outside your door and everything else is contained within a convenient, campus-like setting.
In fact, I found very few properties that could meet all your criteria. However, the one that might best satisfy your needs is the Daufuskie Island Resort & Breathe Spa near Hilton Head Island, South Carolina. It seems to offer the best combination of golf and pet-friendliness in a somewhat upscale atmosphere.
Daufuskie is a special place because it's so remote you actually have to take a ferry from the mainland to reach it (an hour from Savannah or 30 minutes from Hilton Head). There is only one paved road on the island and the preferred mode of transport by locals is by golf cart. The resort has two golf courses, the Jack Nicklaus-designed Melrose and the Weiskopf/Morrish-designed Bloody Point. Both are good, but Melrose is better.
There are multiple types of accommodations at the resort, but pets are allowed only in the Melrose Inn (the main hotel) and the older, non-renovated beachfront cottages, but even these are quite nice as they have multiple bedrooms, full kitchens, dining and living rooms.
The resort's stated policy is that pets must be well behaved and weigh less than 35 pounds, though we saw some there that were overtly larger than that. There is also a $35 per pet/per night surcharge.
Beyond that, I learned that Starwood Hotels are generally pet friendly specifically, its Westin brand (Starwood also owns Sheraton, St. Regis, the W, and others).
Here are a few Starwood hotels in southeastern states with golf on site or nearby. PLEASE NOTE: It's a good idea to call and inquire about each hotel's pet policy because it often differs on a property-by-property basis.
The Westin Savannah Harbor Golf Resort & Spa
Savannah, Georgia
(912) 201-2000
The Westin Resort, Hilton Head Island
Hilton Head Island, South Carolina
(843) 681-4000
Sheraton Myrtle Beach Convention Center Hotel
Myrtle Beach, South Carolina
(843) 918-5000
Sheraton Casino and Hotel
Tunica, Mississippi
(662) 363-4900
Marriott seems to have a few pet-friendly properties, too, but I didn't see any great ones in the Southeast.
Here are some websites that might be useful, too: