I'm a physician, I agree with /u/Ody0genesO that a lot of surgical instruments from Greek and Roman times not only look quite similar to the ones today, in fact, we also kept their names. The Arabs did improve on them, and innovated a lot more like the metallic syringe in the 10th century (see Al-Tasrif). In medieval Europe, barbers and even butchers also doubled as surgeons; which partly explains why physicians still like to rib on their surgical colleagues today, so that's another practice that lives on. ("What is the difference between a surgeon and a carpenter?" -"A carpenter knows more than one antibiotic").
Most of the best practices that survived are quite logical and could have been discovered independently around the world:
Cold sponge (bath) for abating fevers are probably universal, and are still recommended today.
The benefits of exercise have been recognized since ancient times.
Circumcision has mostly been unchanged since BC in some parts of the world, especially the Middle East.
Cauterization: use of very high temperatures to stop bleeding is frequent in many major surgeries.
Opium (to which modern opiates are related) have been used in surgery since BC. The Ancient Egyptians also used it to pacify their kids. I don't know if its use was widespread in medieval times, but the Romans used it during surgery alongside scopolamine (we still use that for treating nausea).
Other drugs: digitalis (as digoxin) is used to treat arrhythmias, aspirin, quinine (from Cinchona, antimalarial first used in South America). Herbs have been used since ancient times, and continue to provide a lot of new pharmacologic compounds. Artemether, now used as a first-line drug for malaria, is derived from qinghao or wormwood (which is used in Traditional Chinese Medicine to treat fevers).
Autopsy: because of the taboo surrounding the dead in many cultures, it wasn't widespread even in the middle ages. However, the Egyptian skill in this arena far surpassed those of the Greeks (who learned it from them, and passed this on to the Romans). The assassination of Caesar was one of the first legal precedents for establishing an official autopsy to determine the cause of death, and obtaining a postmortem report.
Splinting, and setting of fractures. Amputation was also carried out to prevent death from gangrene.
Hirudotherapy (use of leeches in medicine) is making a comeback, but for different reasons other than blood-letting. Blood-letting itself has fallen into disuse and only survives in treating a couple of conditions (hemochromatosis, and polycythemia).
The technique, or the combination of an ailment and its treatment?
Setting of fractured bones and dislocated joints are the only things I can think of where the same problem has the same solution, at least for the simpler cases.
Trepanning, i.e. boring a hole in the skull is used in modern surgery, but apparently dates back to 6500BC - with the difference that the removed part is usually replaced in modern surgery. The technique is used for entirely different reasons, of course.
Blood letting (bleeding) is still in use for hemochromatosis and some other diseases, but as far as I know, is not used for any problem recognised in ancient times.
[Couching of the lens of the eye](http://en.wikipedia.org/wiki/Couching_(ophthalmology)) using a thorn is still used in some African countries to treat cataracts, although this has a success rate of only about 30%.
Absolutely.
Hippocrates's description of how to reduce a dislocated shoulder is still taught today, though often not a first choice anymore. The Hippocratic method of medicine in general, ie taking a history and clinical observation is still the basis of all good modern medicine.
You can go further back than 400 BC for examples, though. The ancient Egyptians left some great records. The Edwin Smith Papyrus dates to 1600BC and describes the treatment of many injuries in detail. It describes suturing of wounds in a manner any modern surgeon would recognise. Though we no longer use meat as a haemostatic agent! In a painting in the tomb of Ipuy, (1300 BC), a physician is reducing a dislocated shoulder, using a similar technique Kocher described in 1870, some 3200 years later. I've used the same method myself in the 21st century. Still works!
Atta HM. Edwin Smith Surgical Papyrus: the oldest known surgical treatise. Am Surg. 1999 Dec;65(12):1190-2
McRae practical fracture treatment 2002
In addition, I expect any surgeon from the modern era, the enlightenment, mediaeval times, classical times and possibly the neolithic would recognise what was being done and why if they were able to observe a modern incision and drainage of an abscess. Probably every damn one of them would find something to nitpick about and claim they'd have done it better. (Unsourced speculation, forgive me this once)
