Or, how to incorporate Pulp in the Cup with D20 rules.
Treat all rolls like oppositional roles, even rolls against a difficulty class (DC).
For rolls against a DC, subtract 10 from the given DC, even if that leaves a negative number. Call this the difficulty adds. So when the player rolls their d20 and adds his bonuses, the GM rolls a 3d6 and adds the difficulty adds. If the player's total beats the GM's total, the player succeeds.
For saving throws against a fixed DC subtract 10 as before, but this time the player rolls a 3d6 and adds their bonuses, and the GM rolls a d20 and adds the difficulty adds to that.
In character versus character contests where both are equally footed (like arm wrestling, foot racing or competing for initiative in combat), everyone rolls a d20.
But in cases where one is acting and the other responding (such as when one is sneaking and the other guarding, or when one is attacking and the other parrying) the initiating player adds their bonuses (or boni) to a d20 roll and the responding player adds their bonuses to a 3d6 roll.
That ought to do it. It will also add some thrill to the game, I think.
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