MB teams are big on buffs. Anyone who can eat their buffs, such as Azure or mifune, will be gold. Even with Root, their mysteries can still be interrupted. It's fairly rare for them to run healers (outside of maybe mabui) and their teams are normally blitz typed - this means if you are running a set-up to outlast them, you will notice a huge damage fall-off around round 3-5. That couples with the fact they just don't run clones (outside of the odd sage naruto clone) means they have little in the way of defense.
As I said before, they don't handle long fights well. Teams with a good healer or two, or someone that shields will throw them off their game. Even though the Earth element is weak to lightning, it was one of my favorite mains to fight back when I ran earth - the playstyle gives most lightning mains a hard time.
MB teams are combo heavy. While this can't be helped much if they are running root (at least for the first couple of rounds), if they aren't then paralysis and chaos are your best friends. Anticipate Danzo and Darui from higher level lightning teams. Darui will cause a ton of paralysis round 2, but danzo is arguably worse since he reguarly causes chaos with his field's passive.
Speaking of fields, gaining field priority in a normal match is important, but against an MB it is huge. Remember the automatic fields are based on your entire team's initiative. While Lightning teams are often speedy, try focusing on your initiative to win that field fight.
While it's not as common in 2.0, don't run a glass cannon team against an MB team. You can't out glass-cannon them. I'm speaking to you, scarlet users.
Now on to the elephant in the room: Root of Warrior and Bloodthirsty. Bloodthirsty is a bit more popular, but not by an overwhelming margin. Bloodthirsty is what allows the MB to go nuts and attack non-stop - it's because they gain a free attack when they kill a unit. We've all seen this at work in our team instances. Learn from it - avoid using * clones as your defense. You can still run clones - Sage naruto's clones and your main's clones are fine. However regular naruto's * clones and wind main's clone army are not. The lightning main will rip through those and keep gaining extra attacks. What's worse, if he runs danzo - he will keep triggering danzo's Ambition passive. 85% of MBs run the high float passive which targets the lowest life ninja on your team. If you run junk clones it will always be them. Just plan ahead and anticipate what will happen if you intend to run them.
Root of the Warrior is the passive everyone is afraid of. There isn't too much that can be done to deal with this. Just bear in mind that ninjas may still have their mysteries interrupted even if they are immune. Also it only lasts two rounds. On the start of round three - the effect falls off. Lastly, the resistance buff is a big deal. If possible, save your pure nin mysteries for round 3.