During the history of Mesopotamia the practice of medicine was considered a sub-specialty of the scribal arts. The goddess concerned with healing was Ninkarrak whose city was Isin, famous for its medical school. Most of the medical texts were written in Akkadian and many of them were found in Assurbanipal's library in Nineveh. Internal evidence points to origins in the Old Babylonian period. Only one text from the earlier Sumer period has been found, but it shows that the Sumerians had quite a large arsenal of medicines used to treat illness.

It was generally thought that illness was caused by angering the gods, by demon possession and the like. To treat this kind of sickness the asipu was called in to pacify the gods or drive away the demon. The asipu was skilled in divination and the interpretation of omens. His treatments included conjurations, magical rituals and incantations.

In contrast, another kind of physician, the asu, attempted cures with truly medical preparations, which included herbals, beer, wine, poultices and oil massage. As an example, a Sumerian tablet from the end of the 3rd millennium BC, probably the first pharmacopeia in the world, prescribes the following:

Pour heated water over a dried and powdered watersnake, the amamashumkaspal-plant, roots of the thorn plant, powdered naga (an alkali-yielding plant), powdered turpentine and faeces of bat. After having lathed the affected area, rub with oil and cover with shaki(?)
As we've seen healing took off into two directions: the asu, dating back to Sumerian times, and the asipu, arising later from Akkadian and Amorite sources. The asu's Materia Medica included:

Social Position

In spite of the number of medical texts that have been discovered, it is doubtful that physicians enjoyed the exalted position they did in either Egypt, Greece or in our own time. Of course, the status of the clients they managed to attract rubbed off on the doctor. Consequently court physicians were held in high esteem and were sometimes sent to foreign courts on special missions. Fees, as now, varied with the status of the client and with the kind of service rendered. Members of the upper class were expected to pay from 5 to 10 shekels while commoners might pay from 3 to 5 shekels. In those days the asipu visited the home of the patients and quite likely also the asu.

Reference

Hawkes, Jacquetta. The first great civilizations. 1973, Knopf, NYC.