Going diceless is remarkably easy. Any time you would roll a die, simply replace it with the number from the following chart.
D20: +11
D12: +7
D10: +6
D8: +5
D6: +4
D4: +3
D3: +2
D2: +1
Thus, if you have a Reflex save of +5, your Reflex save is now 16. If you CMB is normally +18, it is now 29. If your weapon normally deals 2d8 +3, it now deals 13 damage. If your fireball normally deals 10d6, it now deals 40 damage. You should feel free to precalculate and write these numbers on your character sheet to save time.
Example Combat
Blur, Displacement, and Random Tables
Rolls for which there are no possible modifiers should still use the appropriate die. This includes most random encounters tables, random loot tables, random weather tables, displacement, and the confusion effect.
"Roll Twice and Take the Higher Number"
There are a few effects that allow you to roll twice and take the higher number. For d20's, add +3 to the standard result (14). For d12 - d10, add +2. for d8 - d2, add +1.
There are a few effects that allow you to roll twice and take the lower number. For d20's, subtract -3 to the standard result (8). For d12 - d10, subtract -2. for d8 - d3, subtract 1. d2 has no difference.
There are a few effects that allow you to reroll a die. If you are trying to get a higher result, for d20's, add +2 to the standard result (13). For d12 - d8, add +1. For d6 - d2, there is no difference.
There are a few effects that allow you to reroll a die. If you are trying to get a lower result, for d20's, subtract -2 to the standard result (9). For d12 - d8, subtract -1. For d6 - d2, there is no difference.
Optional Rule: Similar Subsequent Saves
There are a few effects which require the same subsequent save to be made over the course of several rounds. The most obvious example is poisons, which you must make the same DC save against until you succeed, but there are also many spells such as hideous laughter (for which you get a second attempt to save) and suffocation. In Pathfinder with dice, these effects often rely on luck to get the weak out.
In these cases, each subsequent save is made at a cumulative +1 bonus to a maximum of 19. Thus, a rogue suffering from a deadly poison would "roll" an 11 on the first round, a 12 on the second round, a 13 on the third round, etc...
Criticals
No dice means no luck, and no luck means no criticals. If you aren't rolling, you obviously can't roll a 20 or a 1. So how do we determine criticals on attacks? There are three methods to consider. My personal recommendation is #2.
Critical Method 1: No Criticals
There aren't any criticals. This goes in line with the general philosophy, but changes the balance of the game a bit. To accommodate for this weapons that have a crit range of 20/x3 or 19-20/x2 gain a +1 bonus to attack. Weapons that have a crit range of 20/x4 or 18-20/x2 gain a +2 to attack. Improved Critical, keen, and similar effects double this.
Unfortunately, effects that occur on criticals just won't happen.
Critical Method 2: Lucky Hits are Criticals
Critical Failure: Missing the target number by one (an attack of 29 against an AC of 30) results in a critical failure (unless you use crit cards or similar, there's no difference between a miss and a critical failure).
Critical Success (20): Hitting the target number on the nose (an attack of 30 against an AC of 30) results in a critical success for weapons with a critical range of 20.
Critical Success (19-20):Hitting the target number on the nose or one above (an attack of 30 or 31 against an AC of 30) results in a critical success for weapons with a critical range of 19-20.
Critical Success (18-20):Hitting the target number on the nose or one or two above (an attack of 30, 31, or 32 against an AC of 30) results in a critical success for weapons with a critical range of 18-20.
In this option, a target cannot be subject to a critical hit more than once a round - all subsequent criticals are just hits. Creatures may choose to forgo their critical for a hit if they think a stronger critical is coming up.
There is some gaming that can be done here. For example, if a barbarian with a crit range 20 weapon is attacking at 32 against AC 30, he can ask his buddy to move out of flanking to get that sweet spot. However, this requires planning, strategy, and knowledge, which should all be rewarded.
Critical Method 3: Powerful Hits are Criticals
Critical Failure: Missing the target number by ten or more results in a critical failure (an attack of 20 or less against AC 30) (unless you use crit cards or similar, there's no difference between a miss and a critical failure).
Critical Success (20): Getting ten or more over the target number results in a critical hit (an attack of 40 or more against AC 30).
Critical Success (19 - 20): Getting nine or more over the target number results in a critical hit (an attack of 39 or more against AC 30).
Critical Success (18 - 20): Getting eight or more over the target number results in a critical hit (an attack of 38 or more against AC 30).
In this option, a target cannot be subject to a critical hit more than once a round - all subsequent criticals are just hits. Creatures may choose to forgo their critical for a hit if they think a stronger critical is coming up.
Whereas criticals are usually an equalizing agent, this places even more power in the hands of those who can hit well, and even less power in the hands of those who can't. It also encourages people not to use the lowest parts of their iterative attacks.
Effects of Going Diceless
Increased Strategy and Teamwork: Consider a game of Pathfinder with dice. Your barbarian moves up to attack the orc. He rolls an 11, adds his attack bonus (+6) for a 17 ... and misses the orc. What does he do next?
If you are like virtually every other player out there, your next turn is going to be the exact same. You've invested into hitting things after all. Maybe it was the roll. You'll stand in about the same spot, roll the same die, and hope for a higher number. Not much strategy there.
With diceless Pathfinder, you know that unless the situation changes, you won't hit the orc. You won't just stand there like a numbskull and swing away again. Instead, you'll look around and try to think of something new and creative.
Perhaps you'll move into flanking for next turn. Perhaps you'll try to trip him. Perhaps you'll make an intimidate check to lower his saves. Perhaps you'll grab a handful of sand and try to blind him. Perhaps you'll convince the bard to aid another and help you get a hit. Perhaps you'll beat a hasty retreat. Whatever you do, it won't be just another swing (and possible miss).
Faster Combat: No dice means combat will go incredibly quickly. There is no calculation to do, and no time spent rolling dice. Turns can be as short as "24 to attack," "That's a hit," "18 damage," with no calculation or dice rolling to slow things down.
Reduced Math: Math is reduced to small modifiers, subtracting damage, and determining whether one number is bigger than another. Gone are the days of 24 + 17 attack rolls, or adding together 13 d8s for a dragon's breath. All of these numbers should be pre-calculated. Going diceless is perfect for those who really hate the math aspects of the game.
No More Eventual Criticals: You'll never hit an enemy with an AC of 100, no matter how much you roll. Likewise, a character attacking at a +100 will never miss. This is not only realistic, it also encourages strategy. You've got to find another way of dealing with the problem than throwing dice at it.
On the flip side, chance can be fun, and rolling high is enjoyable. However, by keeping criticals we can keep most of the thrill of the seemingly random failure or success.