The Battle
I'll divide this into two sections: the opening section and the remainder of battle.
Opening:
For the first third or so of Griffon's life, he is capable of taking to the air at any time and beginning his aerial attacks. However, he always begins battle on the ground - this is where Meteor Lvl 2 comes into its own, if you have it :p
Skip the pre-battle cutscene, wait a second (no longer!), then start charging up a Meteor until it automatically releases - if your timing was right it'll hit Griffon twice for a double hit. This will make Griffon rise up into the air straight away - immediately swivel right and start running over to the right hand side of the stadium. When you get there, you should hopefully see Griffon straight ahead up in the sky. (If you feel daring, at this point you can charge up another Meteor - if it hits, it will do less damage this time, though. On Hard/DMD mode, whilst you can squeeze a Meteor in, it's pretty risky as, if you're too slow in getting it off, Griffon will begin his awful web of lightning attacks).
Anyway, to bring Griffon down from the sky ASAP, plug away with E&I, grenade-rolling, or (fastest of all) the Nightmare Beta. When the cutscene is triggered, this section ends.
If you miss the double hit with Meteor, then evade Griffon's charge and peck, and charge up another Meteor straight away - that should send Griffon rising to the sky :p
If you don't have Meteor Lvl 2, then you'll have to make do with other attacks - Air Raid works quite well, as long as Griffon is on the ground and not up in the air, if memory serves. Otherwise, go to town with DTed grenades, Roundtrips, etc. I'm not 100% on this, but I think DMD Griffon has the same health as Normal/Hard Griffon during this part. At any rate, the opening Meteor Lvl 2 definitely does the same amount of damage in all modes.
Remainder of battle:
Griffon's wing is torn away during the cutscene (which, incidentally, drops 20 red orbs if you want to go and pick them up). He can no longer do his aerial attacks, although he does get a defense boost - he's not as easy to damage as before.
Refer to the Griffon 1 battle section, as everything in there is applicable to Griffon 3. There are a few differences, however.
Firstly, it takes slightly fewer melee attacks to knock Griffon down now (approx 3 Infernoes + several flame kicks and/or Rolling Blazes will do it), but this is offset by the fact that Griffon now has an invulnerabity period when he is knocked over - as long as the blue disc is exposed on his chest, he will be immune to all of your attacks.
It can still be profitable fighting Griffon in this way, however. Use taunting to regain DT whilst Griffon is down, and time the charging of a Meteor or Ifrit punch to hit him after he regains vulnerability. The main advantage of a ruthless knock-down strategy, however, is that it cuts short most of Griffon's lightning attacks. Note that I said "most", not "all."
Secondly, you can get double hits with Meteor Lvl 2. This is impossible to do in the Griffon 1 battle, and insanely hard in the Griffon 2 one. These are far from essential to winning the battle, but are fun to go for :p
If you're interested, then there are various times you can start charging up a Meteor to make it much more likely you will strike for a double hit - one such time is when Griffon hops back (and, at the same time, turns around to face you). He screeches as he does this, so if you start charging a Meteor as he screeches, you will be pretty likely to get a double hit.
A very good opportunity to get a double hit is immediately after jumping left to evade Griffon's futile peck attempt - keep on running forwards for about 4-5 steps, then start charging up a Meteor. This is infallible once you get used to it, unless Griffon is so close to the stands he can't hop and turn around.
Alternatively, start charging up a Meteor immediately after finishing evading any of his lightning attacks. If you're more-or-less in front of him, you can sometimes get a double hit this way - the timing needs to be spot on, though.
And that's pretty much it for the Griffon 3 battle. Evade his attacks; charge up a Meteor or Roundtrip immediately after evading; plug away with E&I or the grenade gun to rebuild the DT.
This time you can get the final hit close up, as a cutscene will be triggered upon Griffon 3's defeat, instead of him flying away like a great big wuss :p
As a final note, some people have complained about the camera in this battle in the past... If you have problems with it, try these counter-measures. 1) Don't use Air Raid. 2) Don't go to the very edges of the stands, onto the stands, or up to the top tier, and 3) Hold R1 to make the camera (slowly) pan around behind you should Griffon disappear out of sight. In my experience, though, Griffon never does his lightning attacks from offscreen - unless you do something like go into the stands. Just stay in the centre of the arena, not to sound didactic about it. There's no profit in going to the outermost edges to fight Griffon.