The Greeks had been impressed with many of the ideas of the Egyptians and borrowed heavily from them at first. Egyptian doctors were often found working in the Greek City States. Of course, the Greeks had a pantheon of their own gods who fulfilled a variety of roles. The story of Pandora's box was used to explain the origin of diseases and medical problems - she opened the box and let everything escape except hope.
The god that would be associated with medicine and healing would be Asclepius. He was supposedly given a magic potion made from the blood of a Gorgon (Medusa) and could use it to heal. The story explains why snakes are still used today as a symbol in medicine. Temples called Asclepions were set up in his honour. Sick and afflicted people would come to the complexes to pray for help in recovering. Theoretically, they would be visited by Asclepius, his two daughters (Hygeia and Panacea) and snakes during the night and they would feel better after a few nights of attention. In actual fact, the complexes often had facilities something akin to modern health spas with baths, saunas, gymnasiums, theatres etc... Therefore, it is not surprising that many people left the Asclepions feeling better than when they had arrived. They had had a few nights rest and recuperation and may have enjoyed the psychosomatic benefits of believing that the gods would actually make them better or perhaps they just let nature take its course but give the credit for time as a healer to the temple of Asclepius.
Greece was an interesting collection of small, competing city states scattered over the Aegean Sea or Eastern Mediterranean. Nature and the elements were essential to the success and livelihoods of most of these city states. It was for this reason that sciences such as maths, navigation and architecture would all make huge leaps as the Greeks tried to keep ahead of nature and one another. This intellectual curiosity spilled over in to the field of medicine as Greek thinkers like Hippocrates tried to explain where disease and illness came from. Unlike praying to a god like Asclepius, Hippocrates used logic to come up with a rational explanation for why people got sick. His theory was known as the Four Humours Theory. It basically said that there were four liquid humours in the body and any imbalance would explain why people were sick. So, too much red in the face might suggest that there was too much blood in the body and so blood letting was carried out. The four humours were lined up with the seasons and elements in a very 'neat' system that sounded convincing. When married to the holistic idea of 'moderation' and the idea of tracking the prognosis of illnesses through writing extensive 'case notes' it is clear why many Greeks were prepared to pay good money to see these so-called experts. Hippocrates even provided his profession with a code of ethics to give customers the confidence to speak freely and trust their doctors. Although the Four Humours Theory was wrong, it is easy to see why it held an appeal and that it established the profession of the Physician as a respected profession in its own right.