Remove turkey neck and giblets; discard large lumps of fat. With poultry shears or a knife, split turkey lengthwise along one side of backbone.
Pull turkey open; place, skin side up, on a flat surface and press firmly, cracking breastbone slightly, until bird lies reasonably flat. Rinse and pat dry. (At this point, you may cover and refrigerate until next day).
Before cooking, squeeze 1 to 2 lime halves and rub over turkey; sprinkle with oregano, then lightly sprinkle with salt and pepper.
Barbecue turkey by indirect heat placing turkey, skin side up, on a lightly greased grill directly above drip pan.
Cover barbecue and adjust dampers as necessary to maintain an even heat.
Cook turkey until a meat thermometer inserted in thickest part of thigh (not touching the bone), registers 185ºF or until meat near thighbone is no longer pink; cut to test (1 1/-2 hours).
Every 30 minutes, squeeze 1 to 2 lime halves and rub over turkey.
Transfer turkey to platter.
To carve, cut off wings and slice breast. Cut off legs and slice meat from thighs.