When you're looking around for free bikes for yourself or for other people in need, it's probably too much to hope for a bike with the latest technologies, like electronic shifting and a frame that fits like a glove. But since plenty of people take up road biking only to find out it's hard work and then promptly let their cycles gather dust in their garages, it's also not out of the question to hope for a bike that will be in good enough condition to get you down the road safely and quickly. You may be able to find a suitable road bike in any number of places -- so long as you're not in a hurry.
Check the "Free" section of the classifieds in your local newspaper or Craigslist or Freecycle.com. If your classifieds have an automatic search function that alerts you when certain things get listed, sign up so you'll be one of the first to find out.
Post your own ad in the "Wanted" section of the local classifieds or Craigslist, letting people know what you want and why you want it. If you're searching for bikes for a good cause, such as giving them to kids in need, include that in your posting and remind them they'll get a tax deduction, if applicable. That may just convince someone that you're the one they should help. Also, post your request on national bike-specific forums, such as Total Bike or Bike Monkey, or city-specific sites for your area, such as Portland Oregon's "Bike Portland" website.
Send a message to your network of family, friends and co-workers letting them know what you're doing. Again, if you're doing it for a good cause, it's worth mentioning.
Visit yard sales, garage sales and estate sales toward the end of the sales day. At the end of the day, people are often much more willing to give things away since they haven't been able to sell them.
Check with local non-profits or cycling centers in your area which may have programs that give bikes to people who don't have them. One good resource to find cycling advocacy groups is USA Cycling's "Cycling Advocacy Organizations" web page.