Dice pools have some nice properties. Chief among them is the lack of swinginess  the number of successes (e.g. the number of d6 that turn up 4+) you roll tends to cluster in the middle, so the wildly extreme results don't come up quite so often.
In Burning Wheel, for example, you might be rolling four dice for your sword skill, with any that turn up 4+ counted as successes. Four dice could generate as much as 4 successes, but there's only a 1 in 16 chance of that  rolling 2 successes is more likely by far.
One of the things you can do with this is use the result as a measure of how awesome you did: the margin of success or failure is a number that's also not too swingy. Bad shit can happen, but it will be rarer than near misses.
If, however, you don't like the aesthetics of rolling 4D6 every time you swing your weapon, or you have a nostalgiainduced allergic reaction to attacking monsters without a d20 in your hand, you've been shut out of this particular statistical bliss. Until now! Aren't you lucky? (At least, as long as you don't mind tables.)
# Dice


d20 
1

2

3

4

5

6

7

8

9

10

1

0

0

0

0

0

1

1

1

2

2

2

0

0

0

1

1

1

2

2

2

3

3

0

0

0

1

1

2

2

2

3

3

4

0

0

1

1

1

2

2

3

3

4

5

0

0

1

1

2

2

3

3

3

4

6

0

1

1

1

2

2

3

3

4

4

7

0

1

1

2

2

2

3

3

4

4

8

0

1

1

2

2

3

3

4

4

5

9

0

1

1

2

2

3

3

4

4

5

10

0

1

1

2

2

3

3

4

4

5

11

1

1

2

2

3

3

4

4

5

5

12

1

1

2

2

3

3

4

4

5

5

13

1

1

2

2

3

3

4

4

5

5

14

1

1

2

2

3

4

4

5

5

6

15

1

1

2

3

3

4

4

5

5

6

16

1

2

2

3

3

4

4

5

6

6

17

1

2

2

3

4

4

5

5

6

6

18

1

2

3

3

4

4

5

6

6

7

19

1

2

3

3

4

5

5

6

7

7

20

1

2

3

4

5

5

6

7

7

8

This table converts d20 rolls into an appropriate number of successes on a dice pool from 110 dice. If you want to roll 5 dice, just grab your d20 and roll  17? 4 successes.
To make this table, I worked out the probabilities for n successes for m dice, and then rounded to the nearest 5% (representing 1/20th of a d20 roll) to see how many times on the d20 n should show up. I handtweaked a couple of entries where the total wasn't 20.
One could use this as a basis for an opposed combat system  Fighters roll d20, treating their level as the # of dice, while monsters use their hit dice. For every point the winner wins by, they do their damage. (So a 6th level fighter duels a 2HD orc, rolling 14 to the orc's 8. The fighter's result is 4, the orc's is 1, for a difference of 3. The fighter comes out unharmed, and the orc takes d8 x 3 hp damage from the fighter's sword.)
It could work similar for saving throws  a fire trap activates with a target of 4. Everyone rolls a reflexes save (maybe that's halflevel for everyone but the thief), taking 5 points of damage for the margin of failure.
Or whatever. :)
To make this table, I worked out the probabilities for n successes for m dice, and then rounded to the nearest 5% (representing 1/20th of a d20 roll) to see how many times on the d20 n should show up. I handtweaked a couple of entries where the total wasn't 20.
One could use this as a basis for an opposed combat system  Fighters roll d20, treating their level as the # of dice, while monsters use their hit dice. For every point the winner wins by, they do their damage. (So a 6th level fighter duels a 2HD orc, rolling 14 to the orc's 8. The fighter's result is 4, the orc's is 1, for a difference of 3. The fighter comes out unharmed, and the orc takes d8 x 3 hp damage from the fighter's sword.)
It could work similar for saving throws  a fire trap activates with a target of 4. Everyone rolls a reflexes save (maybe that's halflevel for everyone but the thief), taking 5 points of damage for the margin of failure.
Or whatever. :)