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Consider running cards that work off of your opponent having a dominating field presence (which they just may, if they work their deck's swarming abilities properly). If you use burn strategies at all, consider running a few Just Desserts (burns 500 for each monster they control=2500 easy damage when they fill their field).
If you tend to lean more towards direct offense yourself, find ways to destroy their monsters. Mirror Force is a clear desire, but Radiant Mirror Force (which is far more affordable) will do the trick just fine against a deck that swarms (it is a just MF, except that it requires your opponent to have 3+ monsters on their side, which will happen easily with Blackwings). A Lightning Vortex would also do wonderfully.
Some Blackwing users also run a Dark Creator (can be special summoned when they have 5+ DARK types in their graveyard, which happens easily), Dark Armed Dragon (can only be special summoned with exactly 3 DARKs in the graveyard; http://yugioh.wikia.com/wiki/Dark_Armed_Dragon ) and also use Vayu (acts as a tuner when it is in the graveyard; http://yugioh.wikia.com/wiki/Blackwing_-_Vayu_the_Emblem_of_Honor ), so you may want to consider a Soul Release (spell that removes 5 cards in graveyard from play) or D.D. Crow (monster that you can discard to remove 1 card in graveyard from play) to keep their graveyards in check.
Aside from that, Blackwing decks that run Bora (which is most of them; Bora is an easy special summon with 1700 ATK and piercing; http://yugioh.wikia.com/wiki/Blackwing_-_Bora_the_Spear ) and Fane (can attack directly; http://yugioh.wikia.com/wiki/Blackwing_-_Fane_the_Steel_Chain ) can be deadly if they also run Sirocco (can take the combined ATK of every Blackwing and put it all on one monster for one turn; http://yugioh.wikia.com/wiki/Sirocco ). Thus, Bora could kill anything and deal lotsa damage if you're defending, with its piercing; or, Fane could attack directly with upwards of 5500 ATK, potentially for an OTK. They could even discard Kalut to increase a Blackwing's ATK by 1400 for one turn ( http://yugioh.wikia.com/wiki/Blackwing_-_Kalut_the_Moon_Shadow ). So, consider running cards that protect you from battle damage. (Defense Draw, Dimension Wall, Magic Cylinder)
Finally, the most powerful Blackwing cards are the Synchros; Armor Master is LV7, can't be destroyed by battle, and can kill almost anything with its effect ( http://yugioh.wikia.com/wiki/Blackwing_Armor_Master ) and Armed Wing has piercing and gains 500 ATK to its base 2300 if it attacks a DEF position monster. ( http://yugioh.wikia.com/wiki/Blackwing_Armed_Wing )
Myself, I run a medium-speed burn/stall; Blackwings are such a common meta these days that I've constructed it in similar fashions to what I've mentioned so as to counter them. The Dark Door is an easy way to limit their beatdown ability, despite their capacity for swarming. Also, if you can stop their first normal summon a turn from occurring with cards like Trap Hole, then their swarming ability for that turn is gone.
There are several Blackwings that I didn't mention; I named the ones that have given me the most trouble. So be wary.
Sources: Been playing since Legend Of Blue Eyes; plays (and beats) Blackwing decks on a regular basis.