Everybody seems to have a different idea about how to do combat maneuvers in old-school D&D (i.e. tripping, blinding, pushing, etc.). This system is sort of stuck together from a bunch of different sources, but I've tried to make them fit elegantly.
Part 1: Getting Damage Dice
Usually you only roll one damage die. There are three main ways to get more than that:
1. Criticals: If you roll a 20, you roll an extra damage die. But furthermore, you can increase your chance of getting a critical by fighting wildly, but this increases your chance of fumbling as well. So if you want to crit on a 15+, then you will fumble on a 6 or less. If you're desperate or just mental, you can go all the way to crit 11+/fumble 10-.
2. Combat advantage: If you have an advantage over your opponent, such as being uphill of them, or flanking*, or they're entangled by animated intestines or whatever, then you get +2 to hit AND you roll an extra damage die.
3. Stunts: If you just want to do something cool, then that's a stunt. Backflipping off the stairs, tipping over the bookshelf, distracting the enemy by pointing over their shoulder, whatever. Just say what you want to do and then we will make up rules for it. But usually the rule will be that you have to make an ability check, and if you suceed then you get to roll an attack and add one damage die if you hit. If you fail your ability check you can't attack this round, and you might suffer some other consequence as well.
Part 2: Trading Damage Dice
To perform a combat maneuver other than just hitting the enemy, you can trade one damage die for an effect. (You have to do this before you roll the damage dice.)
These effects include, but are not limited to: push back (distance = your move rate); knock prone/trip; disarm; stun for one round; etc.
The following effects cost two damage dice: blind; sunder shield/armour; force morale check (requires you to say a suitably terrifying one-liner); etc.
You still have to narrate what you're doing to cause this effect, and it has to be plausible**. If you say "I trade my combat advantage die for the disarm effect" then you get cheetos thrown at you.
I guess this would make it pretty easy to use combat maneuvers, but I'm cool with that. If every important fight starts out with the players trying to blind and entangle their foes, then that's a good thing, right? Hopefully it won't bog down the minor combats too much, because why bother paying 1d8 damage to trip over a kobold when that's almost certainly going to kill him anyway?
I thought about saying that all these rules are available to Fighters only, but that's probably not necessary.
*Still thinking about how to handle flanking without using miniatures. Probably you aren't allowed to just say "Ok, I walk around the enemy so I'm flanking him now," because the enemy will be actively trying to prevent you from doing that. You can get flanking if you've come around by a different path, or if you come up with some cool way of getting past the enemy (i.e. swinging on a chandelier, which would require a DEX check). You could also get flanking if you greatly outnumbered your opponents, but there would need to be some sort of table for that.
**Adjust plausibility to taste. "You can do it if someone could do it in [real life/Game of Thrones/Conan/Star Wars/House of Flying Daggers/Naruto]."