Don't Waste Those Rolls, Use 'em
Presenting: Fortune at the Beginning (FatB). Roll your dice first, and record the results. Then start playing. When a conflict arises, choose a result that will support your desired outcome and use it: "my 20 means that I get to throw you off the building." Cross off results as you use them.
You'll need some kind of control to keep the gamers from just rolling until they get the good numbers they want - otherwise, why bother? So, before the game starts, roll ten dice and record those numbers in a box. You can't use any new numbers until those ten have been used and crossed off.
You could allow the players to keep rolling casually during play, and record those results in another box, up to ten results, and then in another box, and so forth. That allows continuous play and caters to the natural tendency to roll dice whenever, as mentioned. However, it may also push the players to demand lots of petty conflicts in order to use up the "bad' rolls and get to their next pool of results.
The other option would be to have refreshment scenes, as in The Shadow of Yesterday, during which the player gets to roll his next set of ten numbers. That may also have the effect of keeping players from rolling dice the rest of the time, so as not to "waste" any good rolls that might come up.
Before making this post, I asked John whether he knew of any games using FatB. He couldn't name any, but did point me to this four-year-old thread on the Forge where someone asked a similar question (does Fortune at the Beginning exist, and how would it work?). It's worth checking for a little further discussion of the idea and some more thoughts on how a FatB system might work.