Yes, they do. There is a whole series of trans-Europe routes, the Eurovelo network, and some of them are downright crowded with whole families in the summer.

I do some cycling every summer in Serbia and Bulgaria and my route sometimes intersects the Eurovelo 6. It is often a crap shoot if I am going to find accommodation in the smallest towns, because the rooms might all be booked by French and Germans on a cycling holiday along the length of the Danube.