Place Cornish game hens in a large resealable plastic bag.
Mix Greek yogurt, lemon juice, paprika, olive oil, garlic salt, thyme, black pepper, oregano, rosemary, and savory together in a small bowl. Pour yogurt mixture over game hens, turning to coating them completely and get some of the mixture into their cavities. Seal bag, removing as much air as possible, and refrigerate, 2 hours to overnight.
Preheat grill for low heat and lightly oil the grate.
Remove game hens from bags and pat dry. Discard yogurt mixture. Grill until skins starts to brown, about 3 minutes per side. Cover grill and cook game hens, rotating if needed, until an instant-read thermometer inserted into the thickest parts of the thigh, near the bone, reads 165 degrees F (74 degrees C), 15 to 20 minutes.
Remove game hens from the grill and slice open on the breast side, flattening them out like open books. Return to the grill, cut-side down, cover grill, and cook until browned, about 4 minutes.
For a delicious dipping sauce, combine Greek yogurt, fresh dill weed, lemon juice and zest, and some garlic sauce in a bowl. Taste and mix ingredients until you have something that you like. Use this to dip your chicken into while you eat.
Nutrition data for this recipe includes the full amount of marinade ingredients. The actual amount of marinade consumed will vary.
Per Serving: 366 calories;26.1 g fat;
3.1 g carbohydrates;
28.2 g protein;
159 mg cholesterol;
552 mg sodium.