You probably have heard about stories about dying-and-rising gods, like Inanna and Dumuzi and Aphrodite and Adonis. Some examples are also given in this question. Fundamentally,they represent fertility and the seasons. Are there any Asian variants of dying-and-rising gods as well?
This is a deceptively complex question, but Vishnu may be a good place to start.
In terms of the creation/destruction cycle of the universe, Vishnu would die and be reborn an infinite number of times.
However, there is another aspect involving incarnation.
The hero Rama is an incarnation, or avatar, of Vishnu. Rama is born, fulfills his purpose (in this case, defeating the demon king Ravana so that he may be reunited with his wife Sita,) and eventually dies by abandoning his mortal body.
I mention the Rama/Sita story particularly because there is very likely a fertility element involved in their cycle of separation and reunion. "Sita" literally means "furrow", which refers both to the furrows in the earth used for planting seed, and to the the vulva, which itself a type of furrow. Sita is considered to be the daughter of the Earth Goddess Bhūmi, and there are references to a fertility goddess named Sita prior to the Ramayana of Valmiki.
Note that the Persephone story also involves separation and reunification, in this case with her mother Demeter, who is also an earth goddess associated with agriculture. (Sita is likewise said to have reunited with her goddess mother at the end of her incarnation.)
For Westerners interested in reading more about Rama, Sita and Ravana, I highly recommend the William Buck translation, which eliminates the repetitions in the religious text, and makes the core story more accessible to a wider audience.
Most deities in Hindu mythology follow a lifecycle - They come into being, live their lives and then eventually die. They probably don't get sick and "die" in the traditional sense of the word, they just go back to where they came from. The universe itself follows a cycle from birth to death and then birth again. Trinity also follows the cycle of the universe (or multi-verse, as the case may be).
These cycles usually span thousands to hundreds of thousands of years and vary for different gods. The higher you go, the longer the cycle becomes. They're too long to be compared to fertility and seasons and any human related natural/artifical cycles though (for example, Brahma's one day is equal to 10,000 human years, IIRC).