Death is a very large theme in the "The Epic of Gilgamesh." Being that this epic largely represented the Sumerian and Mesopotamians idea I believe the feeling of Gilgamesh himself on death and it's aftermath would be very much the same for most of the society in the time that it was written. Gilgamesh was largely afraid of dying and did everything he could to avoid this inevitable fate.
The first major sign we have of Gilgamesh's fear of dying comes when his friend Enkidu dies. At first Gilgamesh cannot even accept his death, he does not even bury the body until maggots start to appear in Enkidu. Eventually, he realizes that he too must face death one day. This fear is clearly indicated when Gilgamesh states "I am afraid of death" (The Epic of Gilgamesh, 70.) This fear prompts Gilgamesh's trip to the East to see the immortal Uta-napishti, to discover a way to immortality for himself.
This again shows his willingness to fight a dangerous trip to what is considered the end of the world. The scorpion-man advises Gilgamesh of the danger of this trip (The Epic of Gilgamesh 71-73) but this did not stop Gilgamesh for his desire for immortality far exceeded his fear of the dangers of the journey.
Gilgamesh's fear of death again becomes evident when he reaches the sea shore. He is warned by the tavern-keeper that there has never been a way across the ocean and that "the crossing is perilous, it way full of hazard, and midway lie the Waters of Death, blocking the passage forward," (The Epic of Gilgamesh 78.) Again, Gilgamesh presses on for his fear of dying is too great to give up.
The next part of his journey he is instructed to get material to construct...