I've sent in probably close to 100 postcards and have won a set of dream ears.
To enter, send a 3.5" x 5" postcard to:
Disney Dreams Giveaway
PO Box 8629
Elmhurst IL 60126
On the back, put your name, full address, daytime telephone number and birth date.
It's easiest (and cheapest) to get the pre-stamped cards from the USPS - but they are a little too big, so to really comply with the size rule you need to cut them down. Some people have reported that they have won without doing this, but I always do it just in case.
The address for next year's contest will be different - and they've apparently gotten rid of the "postcard" restriction - the new rules say "piece of paper or postcard" with no mention of size.