Call local thrift stores or charity organizations and ask if they will pick up your old furniture. If not, find the closest one and take the furniture there yourself.
Ask the retailer of the new furniture if their delivery driver can haul away your old pieces for a fee. The nice thing about this option is that you swap the old stuff for the new on the same day.
Advertise the old furniture on want ad websites like craigslist. Be sure to include a photo of the pieces to help them sell. Price everything low if you need to move it quickly.
If all else fails, move the furniture to the side of your street or end of your driveway and post a big FREE! sign on it. Unless it's in absolutely terrible condition, it should be gone by the end of the day.