Basically, build a beefy sled to hold your router, making sure that you have more than the width of the stock you're planing on each side of the router. Then you need two supports of exactly the same height for each side of the stock you're planing. Then using a bowl bottom bit or a planing bit (the bigger the better) take light passes. You will probably still need a little bit of sanding when done.
If you use curved supports, you can use the same sort of technique to hollow out shallow chair seats, etc.
American woodworkers woodworking with the router has an example of this jig.
IT IS SLOW AND TAKES A LOT OF PATIENCE-- IT ALSO MAKES A MESS. But if you have complicated grain patterns, etc, the router does a great job.
1 John 1:9