You've got some good advice and some good ideas of your own and no matter what you do with your extra day in Burgundy you won't be disappointed. But since I'm a huge biking enthusiast and spend most of my free time biking in the countryside near Paris I'm going to endorse your own idea of doing a bike ride. There is a great ride, on flat terrain, between Dijon and Beaune that will take you past the vineyards and through some charming villages as well. I've given detailed instructions on this ride to others so I'm just going to copy the same response I've given to others and provide it to you.
First of all, however, whether you are driving or biking you need to get Michelin maps of the scale 1:200,000. You can buy them from their website or once you get to France. The Michelin map number for Burgundy is number 519 (Bourgogne). The Michelin maps have icons for all kinds of historically/touristically interesting things such as châteaux, ruins, churches, abbeys, scenic view points, caves, Roman sites, megaliths, designated scenic roads and many other things. Usually when I'm exploring various regions in France I just look at the map and I am able to plan interesting and scenic drives/bike rides just reading the map. For instance, I usually look for a designated scenic road, which are highlighted in green, and I especially look for towns with the historic church and/or château icon. I also try to make sure the route goes through as many small villages as possible. Usually putting all these things together I find interesting and scenic drives without even knowing where I am going and with no assistance from a guide book. Often these places are never mentioned in guidebooks and remain completely unknown to many tourists. You can buy the Michelin maps from their website and here is a link to the page that shows all of the maps of France: http://tinyurl.com/4bt96ev
I know there are web resources you can use to search for bike paths in the Burgundy region. Let me give you a link to one that I am aware of. http://www.burgundy-by-bike.com/
I also have a list of web resources that show where there are dedicated cycling paths in France. This might already be covered in the link I gave you about biking in Burgundy but since I have this list in a word document it's as easy as cut and paste for me so here it is:
http://www.voiesvertes.com/
http://www.ffct.org/
http://www.af3v.org/Cycle-routes-and-greenways-in.html
http://www.gobicycletouring.info/
http://www.cartovelo.com/
http://randovelo.fr/randovelo/-Anglais-
http://slowtraveladventures.wordpress.com/
http://googleblog.blogspot.com/2010/03/biking-directions-added-to-google-maps.html
Now let's get to my suggested route. I'll describe the route heading south from Dijon to Beaune but you can do it in reverse order too. I have a recent version of the Burgundy Michelin map and it has some of the dedicated bike lanes marked. I'm noticing that there is a bike trail leading out south of Dijon in the direction of Beaune. I'm sure you can check with the tourist office to see where the trail begins in Dijon to take you out of the city. The trail leads out of Dijon and goes through Chenôve and Marsannay and seems to end in Couchey. From here you are going to pick up one of the small white roads, the D122. Keep going south on this road to Vougeot, where you can pass by the château Clos de Vougeot. I've done this road in a car and it is quite nice and goes through several nice, charming villages and a couple of châteaux. From here you want to cross to the eastern side of the D974 to the village of Gilly-lès-Cîteaux. This village lies between the D974 and the A31. Follow the white roads from this town south (between the D974 and A31) to Nuits-Saint-Georges.
Up to this point the route I have laid out is pretty flat and shouldn't present a challenge to pedal. But you've got a choice to make here to avoid biking on the D974, the main route between Dijon and Beaune which is full of traffic. You can head east to Agencourt and then south to Quincey, Antilly, Moux and and Cussigny before ending up in Corgoloin, which is between the D974 and A31. From Corgoloin head south to Ladoix-Serrigny. See the next paragraph for the route from Ladoix-Serrigny to Beaune. The route I just laid out is flat and easy to pedal. I've never done this route so I can't tell you what it offers in terms of scenery but at least you've got the flat factor working for you on this route.
The other route takes you west of the D974 but you've got at least one hill on this route. The first hill you will encounter will be heading out of Nuits-Saint-Georges to Chaux on the D8. The Michelin map shows one gradient arrow on this route, but it is also a designated scenic road with some nice views of the countryside. After Chaux the map shows no gradient arrows on the rest of this route I'm suggesting so any hills (if any) shouldn't be too bad. From Chaux continue to Villers-la-Faye then Magny-les-Villers then Buisson then Ladoix-Serrigny, which is between the D974 and A31. From here follow the white roads south to Beaune, passing through Chorey on the way.
From Beaune you can take the same route back to Dijon, or you can try a different route. If you go back a different route you've basically got to decide on either west of the D974 or east of it. The eastern side is the flat and easy to pedal route. I've never done this route on the east but I bet it's probably not as scenic as the west route, but probably not un-scenic either. If you look at your map just pick a route through the villages that lie between the A31 and D996, following the white roads. Whether you go with the east or west route I think you are going to want to end up in Gevrey-Chambertin and retrace my previous route from here on the D122 back to Dijon. This is the route that will keep you out of traffic.
If you choose the west route it will be more hilly but more scenic. Just look at your map and pick at random which villages you want to go through and aim for Gevrey-Chambertin.
Having said all this perhaps you will discover there is a dedicated bike trail between Dijon and Beaune and you'll stick to that. I know I would follow one of the routes I laid out but I think you know what your comfort level is with a bike better than I do. The route I mentioned is all on small country roads so traffic will be minimal.
So there you go. I know there is also a bike trail south of Beaune that is called the wine trail and if you did a bit of google searching I bet you could find out about that route too. If you ask at the tourist office in Beaune I'm sure they'll tell you all about and probably have maps/brochures outlining the route. Here is the website for the Beaune tourist office:
http://www.ot-beaune.fr/