I think there are two different questions here -- people have known about the existence of major organs for an incredibly long time, long before the Hippocratic school in Greece. And think about what it takes to mummify a human being -- the systematic removal of organs. So that fact that major organs existed (the brain, liver, heart, kidneys, &c) is knowledge that was commonly available. Furthermore, there were plenty of traumatic injuries in the ancient world. The Edwin Smith papyrus (dating to ~2500 BCE) is a lovely surgical resource, and shows a knowledge of the basics of human anatomy. And surgery was practiced routinely in the ancient world. There's a reason the Hippocratic Oath contains, "I will not cut for stone." Coupled with a society far more in line with nature, and a rough understanding of similarity between animals and man, the ancients understood about the existence of organs (at least, the viscera).

Now what I think you're getting at is, did the ancients have any knowledge of anatomy and physiology, which is far different than just knowing about the existence of large organs. That is, did they have an accurate description of organs, their relationship to one another, and their function in the human body. And the answer is, yes -- the ancients had a remarkably good (though still flawed) understanding of human anatomy and physiology. While our education tends to focus on all the things the ancients had wrong about physiology (and don't get me wrong -- there's a lot, like a bicameral, wandering womb), there's actually some remarkable observations, based mostly on skilled observation of living people. Aristotle, we know, dissected animals. This information would be combined with detailed observations of the human body. For example, the Hippocratic "On Glands" is a remarkable description of the lymphatic system, and correctly identifies the spleen as working on concern with the lymph nodes. Galen has many more observations, and he did actually base those of dissections -- though of barbary monkeys. His description of the rete mirabilis (a venous plexus on the brainsteam present in many species, but not in humans) would later lay the ground to move beyond Galenic medicine.

But finally, there WAS actually human dissection practiced in the ancient world, very briefly. Both Herophilus and Eristratus, in Alexandia, dissected human beings (unfortunately, likely vivesection of still alive condemned criminals). This was in the 3rd century BCE in Alexandria. Both these physicians made major contributions to anatomy -- for example, accurately describing the ventricular system in the brain, describing the nervous system, and dissecting out 7 of the cranial nerves. They also described heart valves. It only lasted two generations, but the Eristratists (as the school was later known) was hugely influential in the Roman medical world.