OpenStreetMap
(a collaborative project to create a free editable map of the world ) has a very up to date and accurate cycle route layer available.
I have used it before in Europe to plan large trips.
There are also lots of online add-on services that allow you to use the cycling OpenStreetMap to plan cycling routes and run them either on smart phones or on GPS cycling computers.
Garmin has even used the service to create the specialized maps on their Edge GPS cycling computers.
Some example services include: BikeRouteToaster Garmin Connect ...
more to come (or add them in the comments) As a side note, other proprietary services such as Google maps also have good (and constantly improving) cycling maps.
I however personally find OpenStreetMap is a little more up-to-date and accurate, likely due to the fact anyone can submit a fixes and new pathways.
bicycles.stackexchange.com/users/4534/Rider_X
Google Maps has a Bicycling overlay that can help you.
It does have a few indication that states what type of road you have : Trails
Bicycle-friendly roads
Dedicated lanes
Dirt/unpaved trails
So you might check those dedicated lanes.
An other option would be to use Strava's Global Heatmap where you can see where people are cycling the most.
You can't really see what you want exactly (dedicated lanes), but coupled with Google's map, I find it's a good start to plan a ride.
bicycles.stackexchange.com/users/11462/Bibz
A lot of roads in Germany and Austria are bicycle only.
Usually these are less frequented roads, turned into bike tours.
Then there is a very nice road from Lienz in Austria, to Italy.
http://bicyclegermany.com/ http://hikebikemap.org/ http://www.fahr-radwege.com/ (Austria)
http://www.eurovelo.org/ http://www.suedtirolerland.it/de/rad-mountainbike/fahrradwege-in-suedtirol/
bicycles.stackexchange.com/users/12749/olee22
There is one site that might help you with bike (and other) paths is GPSies (Tracks for Vagabonds) which has a loot of nice features such as: web & mobile site available in 24 languages, metric/imperial system discover routes (supposedly 3,5 mil tracks) (by foot, by wheel, with animals, by motorcycle, on water, in winter...), browse/search recommended routes near your location for different activities (hiking, walking, cycling, mountain biking...), option to create, upload or edit your tracks and it works with following maps: Open Street Maps, Open Cycle Map, Sigma Cycle, Sigma Topo, Hike Bike Map, Google Maps, ESRI)
last but not least are mobile apps for Android and Iphone.
(I couldn't test them because I'm unfortunately on WP :D)
One thing I like is their route creator that seems easier to use than (for example drawing and exporting tracks) on Google Earth.
bicycles.stackexchange.com/users/25208/False Identity
Check out http://trailforks.com/ which is a pinkbike trail platform.
A huge database where people all around the world share their local trails, with short description, photos, videos and exact location and length through google maps.
I live in Greece and it has helped me find some nice trails close to me.
bicycles.stackexchange.com/users/23995/Chris Tsiakoulas
For the Netherlands you might use https://www.route.nl/routeplanner
There (the Netherlands) you can benefit from an extensive bicycle paths network that are marked by signposts thorough the country.
One simply plans a route between two points and marks the knoppunten (knots) numbers along the way.
Then simply follow the signs.
At each knot you find a map with the surrounding, some other routes (with distances between the knots) should you plan to follow a different route.
All routes are bicycle-friendly.
Vast majority is a separate bicycle path.
Sometimes it is a marked bicycle lane.
One might also try the LF-routes ( Landelijke Fietsroutes - country-wide bicycle routes) from https://www.hollandcyclingroutes.com/long-distance-cycle-routes
bicycles.stackexchange.com/users/34697/Mike
Does anyone know of a website that has a map of bicycle paths in Europe that are bike/walkers only, i.e. no cars?
Website with bicycle only paths in Europe?
bicycles.stackexchange.com
<travelling>
bicycles.stackexchange.com/questions/31942
bicycles.stackexchange.com/users/2495/Karlth
OpenStreetMap: There are also lots of online add-on services that allow you to use the cycling OpenStreetMap to plan cycling routes and run them either on smart phones or on GPS cycling computers. As a side note, other proprietary services such as Google maps also have good (and constantly improving) cycling maps. A lot of roads in Germany and Austria are bicycle only. GPSies  (Tracks for Vagabonds) which has a loot of nice features. For the Netherlands you might use  https://www.route.nl/routeplanner. An other option would be to use  Strava's Global Heatmap  where you can see where people are cycling the most. OpenStreetMap, Google Maps and other sites have cycle-specific maps and are often constantly updated and improved. Some European countries have extensive cycling networks with some roads open to bicycles only. These can be explored with dedicated online or offline maps, and are often signposted. Strava's Global Heatmap shows common routes in your area, and other sites like GPSies allow you to create and edit routes or view routes recommended by others.
OpenStreetMap, Google Maps and other sites have cycle-specific maps and are often constantly updated and improved. Some European countries have extensive cycling networks with some roads open to bicycles only. These can be explored with dedicated online or offline maps, and are often signposted. Strava's Global Heatmap shows common routes in your area, and other sites like GPSies allow you to create and edit routes or view routes recommended by others.