I'd like to experiment with it for classic D&D, specifically B/X or LL. While I know there is a D&D 3.5 version out there (from the minds behind E6 if I recall) that I can probably use to do combat, I'd like suggestions on saving throws and moral as player rolls.
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I think it can be done quite easily.
In Labyrinth Lord, each monster has a set of Saving Throws like this:
All you have to do is take that chart, and make a seperate one for monsters, with (22-the original save) as the value of the monster saving throw. Like this:
Whenever there's a call for a saving throw, ask a player to roll a 'Save Breaker'. They want to roll that number or higher. If you don't want them to know what it's for just tell them to roll a d20.
Morale checks are dead simple. Ask the players 'Check if they break' and get the player who looks the least loved to roll 2d6. He'll perk right up. Instead of the GM trying to roll under, the players are trying to meet or beat the morale number (which is more intuitive anyway). If they roll the morale value or higher, the enemies break.
You can maintain the random encounters, you just want to offload the physical dice handling to players. That way you can focus on flipping through pages of tables like a DJ flips through records.
So just do that; ask them to roll whatever matches your random list, but keep your tables behind the screen (as it were). Cluck your tongue and sigh, say "Ooh, I'm glad I'm not responsible for those rolls." Look up the results in your chart and cackle per the original rules.
I hesitate to answer, because the only answer I can think of is tangential to your question.
Really, classic D&D relies on the DM is making certain kinds of decisions, and part of that decision process involves rolling to decide things behind the scenes. I absolutely love semi-diceless mechanics, but there are so many parts of class D&D that I just can't think of any way to put the dice in the players' hands.
For example, assuming random encounter tables are part of classic D&D, how would this be turned into a player roll? I can't think of any character action that could be meaningfully made to serve the same purpose. The alternative to making it character action resolution is to just hand the die to the players and say, "roll, and tell me what is says," which doesn't nearly harness the awesomeness of semi-diceless mechanics.
Similarly with saving throws and morale checks for monsters. I can imagine saving throws becoming a "overcome resistance" roll for the players. However, that would quickly start feeling like "rolling to see if the spell works" and give a very different feel to D&D magic.
With morale checks, I can imagine an "intimidation check", but that would take away much of the surprise and uncertainty when monsters retreat: are they running? retreating? regrouping to return soon? faking a morale break to lure the PCs into an ambush?
So, I'd say that classic D&D is not well designed to hack-in semi-diceless mechanics. It could either be done superficially by just handing the players the dice when the DM would normally roll, but that wouldn't gain much. It could be done more meaningfully, but at some point that wouldn't feel like classic D&D anymore, and at that point you might as well choose a system that does it natively.
As an example of a "D&D-like" game that is based on a semi-diceless system (although it does retain DM rolling on random tables as an option to respond to a player's failed roll), you might take a look at Dungeon World.