I was debating whether to just comment on elemtilas' fine answer, but I think I have enough to be worth an answer on its own.
TL;DR: super-tough blood vessels would be very useful, and just cos you didn't die immediately doesn't mean you're going to live til next week.
Once you've excluded the head, immediately fatal or disabling injuries from blunt force trauma are difficult to inflict, unless you use a lot of force (like, driving a truck into someone). You might break the spine (which is awkard, in combat, unless your opponent is distracted, disabled, or unaware of you) but that's been covered well enough in other answers so I won't repeat it.
That leaves you with "rapidly fatal injuries" which will kill you or render you helpless in a couple of minutes and "eventually fatal injuries" which won't stop the fight but will probably stop you getting a rematch tomorrow. There are also "crippling injuries" that will take you out of the fight, letting your opponent finish you off. I won't consider those right now, though.
So I'm gonna cheat a little bit and say that the next most vulnerable thing you need to worry about is "the circulatory system" (or at least, the major arteries) as the big one, followed by either "the lungs" (which I guess means the bits of your chest which cover them) or "the long bones" (eg upper arms and legs) but they kinda tie into the circulatory system with regards to how quickly you can die from an injury to them.
Here are some important observations, which may seem obvious but should be born in mind anyway:
- Blunt trauma isn't just caused by weapons. I pick you up and slam you on the floor, that's blunt force trauma. Indeed, if you've got indestructable hands, feet and head, wrestling or jiujitsu isn't a bad means of disabling or killing you...
- Just because it is blunt trauma doesn't mean that there isn't going to be holes and external bleeding. Open fractures can have jagged ends of bones sticking out through the flesh (or just jagged holes, if the bone is pulled back inside).
- Just because your hands and head are indestructible, doesn't mean that hitting them is pointless. The head is still a nice lever; a strong blow to the side can knock you over, a strong impact to the top can damage the thoracic spine. A strong enough impact to your hands and feet can still break the bones that attach the to the rest of your body
Death by haemorrhage. Even without an open fracture letting all your juices ooze out, internal bleeding can arise simply from blood vessels being crushed open against bones, or the jagged ends of closed fractures severing them. You can lose a litre of blood internally to broken upper arm bones, and two litres to broken upper leg bones. Serious trauma to the chest, abdomen or pelvis can easily cause enough internal blood loss to kill you. Hypovolemic shock can put you out of the fight in minutes (though be fair, breaking a big bone will probably render you pretty helpless immediately), and potentially kill you in minutes to hours.
Finally, there's the big one: aortic rupture. This is what will happen to you following one of those super-strength wrestling slams. You might not die instantly if your aorta become unhitched, but you'll be gone within a minute.
Death by asphyxiation. Break enough ribs, and you'll have a problem filling your lungs, and you won't be fighting at your best if you can't breathe. If one of those rubs punctures a lung, your chest cavity can end up filling with air, causing pneumothorax (which is disabling) and eventually fatal tension pneumothorax. Maybe you didn't puncture a lung, but merely tore a blood vessel or two? Haemothorax. Maybe you get both! Haemopneumothorax!
Oh, and if you survived one of those long bone breaks, maybe you'll get a fat embolism. I'm not sure if "indestructable head" means you're immune to strokes, but that won't save you from a pulmonary embolism which may or may not kill you, but you certainly won't be at your best if big regions of your lungs have become non-functional.
There's also lots of other fun to be had with regards to disabling (nerves aren't very crush proof) and eventually fatal injuries (organs go pop, peritonitis is a nasty way to go, and then there's rhabdomyolysis...), but I think this should be enough to keep you going for now.
PS: I'd be looking at entangling, blinding, burning or electrical weapons meself, if I weren't allowed projectiles, but that sounds like a topic for another day.