It is a commonly agreed upon lie that you cannot see the European countryside by train. The European rail system covers much more ground than you might expect if you didn't have proper railway maps at your fingertips.
The internet has exacerbated the problem because comprehensive rail maps are difficult to display in the limited bandwidth and limited pixel width of the common web page.
So, if you'd like to explore some of France's most compelling countryside while leaving the driving to somebody else, follow along and you'll be able to plan your own personal itinerary to France's Burgundy (Bourgogne) region, specifically, the Cote d'or, the gold coast.
Burgundy Starting at Paris
If you're flying into Paris you can take the TGV directly from Charles DeGaul Airport (CDG) to Dijon or Beaune. Dijon is a town in northern Burgundy with a fine Medieval core and very interesting market. You should stay a couple of nights in Dijon and let the jet lag dissipate if you're just in from the US.
You can also start your journey in Vezelay, a hill town developed to protect Vézelay Abbey; the abbey church and the town make up a UNESCO World Heritage Site. The town is known for its wine and cuisine.
Getting into the Burgundy Countryside by Train
From Dijon (the station is called Dijon Ville), you can take a short hop to either Beaune or Chagny (Beaune has a TGV station as well). If you are traveling on a limited day Eurail pass, don't use a day of it for short trips like Dijon to Beaune.
Beaune is a fine place to make a hub for your Burgundy visit. Stop by the Beaune tourist information tourist office on the ring road at 6 Bd Perpreuil and check out your options. If you like walking (and there is no better way to survey the stunning vineyard landscape than strolling), the office can offer you a packet of walks with maps and directions. You can also check out the Pass Beaune Comme Bourgogne, which will give you discounts on touring the sites--don't miss the Hôtel-Dieu (Hospices de Beaune), which combines community charity with the wine industry in a very interesting way.
There are also alternatives like tours that the tourist office can advise you on.
There is also a bike route from Beaune to Santenay. The brochure is a PDF download and only in French. Bicycles can be rented in Beaune, inquire at the Tourist Information Office.
An alternative to a stop at Beaune involves staying on the local train a little longer and getting off at the village of Chagny. Here you'll want to stay at the Hotel Restaurant Lameloise, 36 Place D'armes, which has a top restaurant despite being just a three-star hotel.
Further down the train line is Chalon-sur-Saône. Here you're in the heart of the Côte Chalonnaise, with famous appellations like les Mercurey, Rully, and Montagny.
Here you can catch the Voie Verte, a paved path that runs 117km away from roads and cars using old railway and "haulage" paths. You can walk or ride a bike on it through the vineyards of Burgundy. Here is a map and information on France's first voie verte.
Consult the Tourist Office at 4 place du Port Villiers.
Train Resources for Burgundy
Continuing South from Burgundy
Just south on the rail line is France's culinary capital Lyon. Travel further south to discover the wines and towns of the Rhone Valley.