In this case, Western culture is the outlier. As Simon explained in the comments, serpents feature prominently in many non-western mythologies:
In Australia the Aboriginals believe that a Rainbow Serpent created the landscape, embodied the spirit of the fresh water and punished lawbreakers. In Cambodian Mythology, they believe the Khmer People to have been born out of union of Indian and Nagas or Serpents. For Hindus it represents rebirth due to the casting off of it's skin and being 'reborn'. Search for Nāgas for more Hindu myths, they feature prominently in several myths. Koreans fear the snake and worship it as a god. In Mesoamerica we have Quetzalcoatl who was a 'feathered serpent' patron of arts, crafts and merchants.
Why is this? To answer that question, I'm going to take a look at the epic of Gilgamesh, which is from a similar region as the bible. Here's a description of a snake from the epic:
'This plant, Ur-shanabi, is the "Plant of Heartbeat",
with it a man can regain his vigour.
To Uruk-the-Sheepfold 1 will take it,
to an ancient will feed some and put the plant to the test!
'Its name shall be "Old Man Grown Young",
I will eat it myself, and be again as I was in my youth!'
At twenty leagues they broke bread,
at thirty leagues they stopped for the night.
Gilgamesh found a pool whose water was cool,
down he went into it, to bathe in the water.
Of the plant's fragrance a snake caught scent,
came up [in silence], and bore the plant off.
As it turned away it sloughed its skin.
Then Gilgamesh sat down and wept,
(Epic of Gilgamesh)
To summarize: Gilgamesh has a plant that grants him immortality, but the plant is taken from him by a snake. The snake then becomes immortal, as snakes can "shed their skin" and thus be reborn.
I'm not trying to suggest that Gilgamesh is a precursor to the bible, or that Gilgamesh represents every other myth that features serpents, but it is illuminating as an example of how other cultures viewed serpents. It's clear from the epic of Gilgamesh that snakes are strongly associated with rebirth, immortality, and life. I hope that goes some way towards explaining why serpents aren't portrayed as "evil" in other cultures.