Marinate the game hens in the fridge overnight. Remove the birds from the brine and pat dry with paper towels. Smoke the Cornish game hens at 225 degrees Fahrenheit for 3 to 4 hours, or until the thigh temperature reaches 175 degrees.Allow the birds to rest for half an hour after smoking before splitting through the breasts and serving.emove the birds from the brine and pat dry with paper towels. Smoke the Cornish game hens at 225 degrees Fahrenheit for 3 to 4 hours, or until the thigh temperature reaches 175 degrees. Allow the birds to rest for half an hour after smoking before splitting through the breasts and serving.