DIAGNOSIS That's coil whine from the internal power components, usually the capacitors or inductors. It's an annoying but common electronic thing, not typically a sign of imminent failure.
FIX First, check the power supply. Make sure the adapter is plugged firmly into both the wall and the router. Try a different wall outlet on a different circuit if possible. Sometimes dirty or fluctuating power can exacerbate the noise. If the noise changes pitch when the router is under heavy load (big file transfer), that's classic coil whine.
IF THAT DIDN'T WORK There's not a user-serviceable fix for internal coil whine. If the noise is very loud and the router is new, it might be worth contacting Linksys support about a possible replacement under warranty, as it could indicate a marginal component.
HEADS UP Don't open the router to try and fix this—you'll void the warranty and risk shock even when unplugged, as capacitors can hold a charge. If it's just a faint buzz and everything works, it's more of an annoyance than a functional problem.