I've thought about a system where all GM-facing rolls would use a flat value and all player-facing rolls would include a, er, roll.
How this would work is that monsters would have flat save/DC values that players would need to hit in order to cause an effect - AC of 15, Fort of 12, Reflex of 13, etc. The GM wouldn't need to roll for monster attacks, either - everything, including weapon attacks, would require a roll for the player.
If a goblin shoots you with a shortbow, instead of the GM rolling an attack roll the player would roll their Armor save, similarly to how you'd roll a Fort, Reflex or Will save. Roll well enough to beat the DC and the attack misses, roll low and it hits.
I like the idea because it leaves all the rolling in the hands of the players, and simplifies the DM's job since they don't need to make rolls for their monsters - it's simply a flat value that the player has to roll against instead. It also preserves and even enhances the fun the player has, where they get to roll dice and see if their character gets out of the way in time or not. The downside is that it definitely, 100% sets up GM controlled characters and player controlled characters with different rules, and I know some people hate that.
4e-style spell attack rolls versus set saves, similar to AC, is also a good solution.