AWESOME awesome question. This could make for a great science lesson.
The first scientific principle at hand to consider is terminal velocity. In simple terms, terminal velocity is how fast an object can fall to the earth. I found a great website that talks about terminal velocity, and even computes the terminal velocity of different sized raindrops we find in nature, as well as explains why clouds float even though they're big blobs of raindrops.
Here's that link.
So, if you assume you have a large raindrop, it reaches 20 miles per hour in it's journey to earth. The next thing you need to know is how high up a cloud is.
This site lists the different types of clouds and their heights. According to this site, most rain clouds are around 6,000 feet up, or around a mile high.
So if a raindrop falls at a rate of 20 miles per hour, and it starts it's journey a mile up in the air you have a raindrop taking 1/20th of an hour to hit the ground, or in other words, about 3 minutes.
Of course, that's assuming a large raindrop and a typical rain cloud. Drizzle could take up to 45 minutes to hit the earth! As always, your mileage may vary.