Depending on the killstreak, they can be destroyed in a variety of ways. Many flying killstreak rewards, such as Harriers, Attack Helicopters, Pave Lows, AC-130s and Chopper Gunners can all be taken down with AT4-HS, Stinger and Javelin Missle launchers with relative ease. However, many of these aircraft will deploy flares to counter the lock on feature of these weapon. Generally, the higher the killstreak, the more sets of flares the aircraft will release. Harrier's and Attack Helicopters release none, Pave Low's release 1, Chopper Gunner's will release 2 and AC-130s will release 3.
Other ways to destroy killstreak rewards include activating an EMP, calling in a Harrier, which will attempt to shoot down other killstreak rewards and standing close to reward and firing off a Stinger Rocket. If you are close enough, the aircraft will not have any time to release flares and will be destroyed instantly.
Sentry Guns can be killed by anything causing damage, but most effectively by knives and semtex.
UAVs, Helicopters and other airborne killstreaks can be shot down with bullets, launchers, SAM Turrets and on MW2 the Harrier Strike as well. Predator Missiles can be destroyed in MW3 through the use of a Trophy System. SAM Turrets, Sentry Guns and Remote Sentries can be destroyed by knifing or lots of damage done by bullets. Assault Drones can be temporalily disabled through flashbangs but a well placed EMP grenade makes it a pile of junk afterwards. Recon Drones can be shot down by anything, knifed if low enough and sometimes can be blown up by stray bullets from Sentry Guns and Assault Drones. Airdrop traps can be destroyed by triggering it and quickly running around a corner for cover. Care Packages cannot be destroyed but they can be changed into Airdrop traps through use of Hacker Pro in Black Ops. Stealth Bombers can be destroyed by a well timed EMP. EMPs can be countered through use of Assassin. Juggernauts can be killed easily by having a team of three people continuously firing at the Juggernaut's torso or head. Attack Dogs can be countered by shooting a few bullets at them (They have to hit the dog). Rolling Thunder cannot be shot down but getting into a building might mean the difference between life and death. Tactical Nukes and M.O.A.B.s cannot be countered in anyway possible. Even killing the person who deployed won't help either. The M.O.A.B.'s fallout can be countered by Assassin due to the EMP effect from a M.O.A.B. but death is inevitable when an enemy M.O.A.B. is incoming. At that point when it is announced just try to take out as many enemies as you can within that small time. AC-130's take two missiles to take down. The first one to get rid of it's flares and the second missile to swat it out of the sky. Quite a lot of killstreak-counters aren't there?