We will have 5 or 6 days in Germany and want to see all of these things. Does anybody have any tips on how to travel this route (or something close to it) efficiently?
5-6 days is not enough for all 3. I would suggest dropping one of them. Otherwise, you'll spend too much time travelling from place to place and not actually seeing anything. Which area you drop will determine the best mode of travel. The Romantic Road area is best done by car. The Rhine/Mosel River area can easily be done by car or by train/bus (a little easier by car). Going from Munich to the Rhine would probably be best done by train.
In my opinion, if the "Romantic Road" did not connect two places people want to go between, no one would ever drive it. Yeah, there are some places of interest on the road, but they aren't worth the trouble of getting to them, particularly if you only stop briefly.
Driving the RR without stopping will take about as long as going around it by train, and the gas alone will cost more than a Bayern-Ticket for the train (not to mention a day's car rental).
I also agree with the previous poster that the time involved with getting to/from the middle Rhein is not worth it.
What happened to everything else around Munich (Salzburg, Berchtesgaden, Eagles' nest, Garmisch, Innsbruck, Nürnberg, Würzburg) and Munich itself? Spend your time efficiently in the local area and save the Rhein for another day.