Just to get the question out of the way up front: Nastya's teddy, found on Volga, does not count as the teddy bear mentioned in the achievement text. After picking up Nastya's teddy, the achievement tracker did not increase.
Also be aware that like the other collectibles, once you've acquired them, they remain in your inventory. So if you die on a level after collecting one and have to re-start, you do not have to re-acquire the collectible [thanks to NoHeroes94 for the info].
Fish - On VOLGA, at the end of the mission where you secure the railcar, in the side room where you can lure and kill the Tsar fish. If you turn left after walking in, there'll be a boat standing against the wall with a fish hanging down on the right side.
Sun - On CASPIAN, there is a derelict ship that has a hermit who lives inside of it. It is located here:
Be aware that driving to it is risky, so watch out for things that will get your vehicle stuck and avoid those flaming pools of oil at all costs. The ship itself is infested with spiders, so prepare for that (the Tikhar's fire rounds make short work of spiders).
Just below the level where the hermit is sitting, you'll come across a shelf with a number of items including a tape recorder. If you're looking at the tape recorder and turn right, you'll spot the sun dangling from a shelf.
Bear - On TAIGA, near the beginning of the level, you find a schoolhouse. If you get to the broken bridge where the group of guys drop out of a tree and address you, you've just passed it.
After entering the schoolhouse, turn right and go through the open doorway into the next room. The toy will be in the left-hand side of the room dangling from the window frame just above an actual teddy bear like Nastya's [Thanks to Ix x LEMON x xI for providing this early location].