If I steal a gear with Akshan, and Akshan dies, do my opponents get their gear back?

Yes, when your Akshan leaves the board, you lose control of the gear, and control reverts to its original owner.

According to the ruling in RiftJudge FAQ #5396 and RiftJudge FAQ #9043:

1. Control-change effect: Akshan’s ability creates a control-change effect that lasts only for as long as that specific Akshan remains on the board.
2. Akshan leaves the board: Once Akshan leaves the board, that control-change effect stops applying.
3. Recalculation: When your control-change effect stops, the game recalculates control. If no other control-change effects are active on that gear, control automatically reverts to the original owner (the player who brought the card into the game).

Important Nuance: As noted in RiftJudge FAQ #9043, if the gear was attached to a unit, it does not automatically detach or return to the opponent's base when control reverts; it remains attached to its current unit until it is moved by another card effect or the unit it is attached to leaves the board. However, because you no longer control the gear, you cannot use any of its activated abilities (as you are no longer its controller).