Travel in March

You can see actual day-by-day temperatures over the last 20 years for many cities on wunderground.com. Choose Historical, then Monthly.

Until March 21st it's still technically a winter, sometimes in reality too.

I was there in March, in the Alps of southern Bavaria, about 15 years ago. Skiing was being washed out by rain, but it was still cold. I was there, farther north, three years later, and it was chilly (I think we had a trace of snow one night), but the weather was basically good. I was on the Rhein for four days. It rained one of those days.

I don't know if he still does it, but Rick used to give a chart of average temperature and daily average rainfall for various parts of Germany in his Germany guidebook. All I remember is that October is the driest month in Bavaria.

We were in Vienna, Budapest, Berlin, Munich in March 2017 -- Germany at the second half of the month. Raincoats and fleece jackets. The first week (Vienna) was cold enough for leggings under jeans. Crowds at Neuschwannstein on the weekend (Disneyland-like) but the rest was manageable.

That part of Europe is about as predictable as Chicago or New York - in other words, it could be anything - early warmth, late snows. Take into account that sunset's around 6.15 and if it's not sunny, even during the day, the countryside won't be very scenic.

"We fly into Frankfurt and we’re a little unsure of the weather."

Historically, Frankfurt has the least precipitation, on average, in the months of February, March and April:
http://www.worldclimate.com/cgi-bin/data.pl?ref=N50E008+2100+1064001G1

I carry a folding umbrella no matter the month. Take warm clothes for March.

"We would love to see the countryside and villages and castles."

Spring comes later than early March. But near Frankfurt, the Rhine Valley, with its castles and villages, is still very impressive, even without the greenery. Marksburg, the Rhine's most impressive castle tour, is open year-round. It's in Braubach, one of several fine old-world, riverfront towns that lie between Bingen and Koblenz. Map: http://www.mittelrhein-wein.com/Bilder/rheinlauf-gross.jpg

http://www.marksburg.de/en/
https://www.romantischer-rhein.de/uploads/pics/Altstadt_02.jpg

I would probably choose to stay in a larger town like Mainz or Koblenz during March, places where there is something to do after dark, and take day trips to the villages and castles.

You'll be fine, just dress warmly. This web site has near complete travel weather:
for March:
https://weatherspark.com/m/148124/3/Average-Weather-in-March-at-Frankfurt-Airport-Germany
for all months, including a section on the best time of year to visit:
https://weatherspark.com/y/148124/Average-Weather-at-Frankfurt-Airport-Germany-Year-Round

I wouldn't want to be as discouraging as some of the other posters. Yes, you will have to be prepared for all kinds of weather. Yes, you will have to take warm clothes and an umbrella or a rain coat. No, you may not expect summery temperatures.

That said, you may encounter some absolutely beautiful early spring days. The Rhine Valley usually has spring somewhat earlier than in other regions, so if you want to see "countryside and villages and castles", that's where you want to go. And March may even have sunny days and somewhat stable weather conditions and doesn't have the reputation of being such a wet month.

If you are into flowers, you may want to look for botanical gardens, renaissance gardens, or parks that are famous for their crocus bloom. Or go hiking in some deciduous forest and enjoy the bloom of the wood anemone (Anemone nemorosa). It's probably a bit early for tulips and for orchards to bloom, but you may be lucky.