I will be listing mons under each category from top to bottom in what I believe to be viability/strength as a counter/check.
Counters
None.
Strong Checks
Munchlax deals with most Murkrow sets to a decent extent. Using a 31/14 HP and defensive spread, you can easily live Brave Birds with Berry Juice, and threaten to OHKO back with Body Slam (if LO) or Ice Punch, or even stall out Bbird/LO's recoil with Recycle. Recycle makes sure that you stay healthy throughout all Murkrow encounters. The great thing about Munchlax's BJ set is that it has a high HP pool and low defense, so Murkrow basically kills itself with Brave Bird. Other Murkrow sets can't really threaten Munch much either aside from paralyzing it. Munchlax also doesn't care about getting paralyzed which is a bonus.
Archen can switch in to non LO Krow very safely, as nothing 2HKOs it aside from like HP Ice. It can threaten to kill back with Rock Slide even through Defaitist thanks to its ridiculous attack stat. It can easily Roost off any damage that it takes after switching in and can force Krow out.
Chinchou is an obvious one. Being the bulkiest Electric type that isn't weak to any of Murkrow's common moves, its a great switch in to many Krow sets. With Eviolite or Berry Juice, you can easily tank BBirds, but you have to be careful for moves like Sucker Punch. So when Chinchou is used with the purpose of checking Krow, either RestTalk or Thunderwave should be used to avoid Sucker. It's also immune to the ever so common Thunderwave Murkrow.
Tirtouga is one of the best Rock types in Little Cup. Thanks to its very impressive base defense stat, it doesn't take much from any of Krow's physical attacks. Bulky Tirtouga can swap into most Murkrow sets. Most Murkrows opt for Heat Wave and Dark Pulse/Icy Wind for their special coverage, leaving Tirtouga safe from HP Grass. It can then set up SR, threaten to KO with Rock Slide, AJ, or Scald. Dark Pulse however, is very threatening to BJ Tirtouga. Brave Bird + Dark Pulse from BJ Murkrow is very likely to 2HKO BJ Tirtouga. With careful play however, one can easily whittle LO Krow out of its HP pool using Tirtouga as a pivot.
84 SpA Life Orb Murkrow Dark Pulse vs. 0 HP / 76 SpD Tirtouga: 16-19 (76.1 - 90.4%) -- guaranteed 2HKO
84 SpA Murkrow Dark Pulse vs. 0 HP / 76 SpD Tirtouga: 12-15 (57.1 - 71.4%) -- guaranteed 2HKO
Growlith is an unexpected check. With Intimidate and 25/14 defenses, it can even shrug off 2 LO Bbirds so long as Stealth Rock isn't up. Flamethrower, WoW, or Toxic all hurt or cripple Murkrow significantly. Murkrow also fails to 2HKO Growlith through Stealth Rocks if Murkrow has no Life Orb.
-1 236 Atk Life Orb Murkrow Brave Bird vs. 236 HP / 156+ Def Eviolite Growlithe: 9-13 (36 - 52%) -- 11.3% chance to 2HKO
-1 236 Atk Murkrow Brave Bird vs. 236 HP / 156+ Def Eviolite Growlithe: 7-10 (28 - 40%) -- 11.3% chance to 2HKO after Stealth Rock
General checks
Lickitung counters almost all Murkrow sets except for ones like Band and Calm Mind which are fairy rare. With 27 HP and 18 defense, it is rarely 2HKO by Life Orb Bbird with no hazards, which is amazing considering Lickitung is neutral to it. However, Lickitung has almost zero offensive presence and is generally outclassed in most other scenarios by Spritzee due to its superior typing and offensive presence.
Spritzee can shrug off non LO Brave Birds, and heal back with Wish while always being able to OHKO back with Moonblast. It also resists Murkrow's Dark Pulse and is neutral to Heat Wave. However, it's almost always 2HKO by LO Bbird, keeping it from being a counter/strong check. Spritzee is still 2HKO by Bbird sometimes after Stealth Rocks. Spritzee can also Aromatherapy paralysis if needed.
236 Atk Murkrow Brave Bird vs. 212 HP / 196+ Def Eviolite Spritzee: 10-13 (37 - 48.1%) -- 56.3% chance to 2HKO after Stealth Rock
Vullaby and Lileep are other excellent choices. Brave Bird cannot 2HKO Vullaby without SR or Life Orb, while Vullaby's Brave Bird hurts back a lot more. Vullaby is also the only check with access to U Turn, meaning you can gain momentum when Murkrow roosts or switches out. Lileep threatens Murkrow with Ancient Power and/or Toxic, while stalling with Recover.
Slowpoke is in the same boat as Spritzee. Strong physical defenses let it tank non LO Bbirds, but Slowpoke is a bit shakier due to Dark Pulse and not being able to OHKO. Slowpoke does get Regenerator though, meaning it doesn't have to rely solely on Wish for recovery.
Hippopotas similar physical bulk to Lickitung, however it's a rather poor choice this generation due to Sand Stream nerfs. HP Ice and Grass also 2HKO Hippo. Hippo can't do much back aside from Toxic or Whirlwind it.
Swirlix can swap into LO Krow with the right spread, heal with Berry Juice, and force Krow out or kill it. 24 HP and 14 defense are the needed EVs to reliably survive Krow with no hazards, so this spread is generally best used for Cotton Guard sets. Unfortunately, it can only switch in once, and extremely hinders Swirlix's sweeping abilities due to not being able to set up fully after losing Berry Juice.
Most rock and steel types can check Murkrow to a certain extent. Aron, Nosepass, Tyrunt, Magnemite, Pawniard, Amaura, Onix, Geodude, etc. However most of them are shaky shaky checks, as most Murkrows carry Heat Wave, and many Rock types being vulnerable to Icy Wind or HP Ice. Many of these mons are also not horribly viable, such as Geodude and Tyrunt (somehow this shitter is C rank lol). Aron and Amaura are the most notable here, having uses outside of checking Murkrow and almost always beat it 1 v 1, even when switching into a move. Aron's unique typing in LC gives it neutrality to Heat Wave and x4 resistance to Brave Bird, letting it switch into anything and threaten to OHKO back. Amaura's Nature Power combined with Refrigerate makes Murkrow's Sucker Punch useless, and along with its Rock typing and decent base stats it can switch into Brave Bird twice, making it a decent check/revenger.
Revengers
Amaura is already listed under the checks section, but its so damn good of a revenger that I'll put at at the top spot for revengers. As stated before, Nature Power Refridgerate's combination allow it to bypass Sucker Punch and most of the time, OHKO, which is a unique trait for scarfers.
Carvanha can outspeed Murkrow after +1 from Speed Boost and threaten with Waterfall or Ice Beam. It doesn't take too much from Sucker thanks to its Dark typing. +1 speed Aqua Jet is also an obvious choice, bypassing Sucker Punch's slower priority.
Meditite and Meowth's combination of Fake Out + Additional Priority make them great choices to revenge Krow with. Fake Out with Life Orb from either Meditite or Meowth both bypass Sucker Punch and deal damage of around 25%-60% depending on Murkrow's nature and item. Bullet Punch and Feint then do additional damage, bypassing Brave Bird for Bullet Punch, and Feint bypassing Bbird, Sucker, AND Prankster Roost. Meowth is kind of a meh pick overall, but being able to 2HKO LO Murkrow without taking any damage is pretty cool and unique.
Eviolite Elekid doesn't get OHKO'd or anything by Sucker Punch, and OHKOs with Volt Switch/Thunderbolt. Its almost unique 20 speed allow it to outspeed everything that isn't priority.
Other revengers include basically everything with a Choice Scarf that isn't weak to Sucker, or anything with decent priority. Snover, Murkrow, Pawniard, Chinchou, Timburr, and Croagunk get special mention for being decent at accomplishing a revenge.
Other options aside from revenging include setting up on Krow. SS Tirtouga accomplishes this nicely, being able to dish heavy damage after Smashing. CG Swirlix with a physically bulky spread easily sets up on LO Krow and heals back with Draining Kiss.
