Netherlands in March - suggestions for non-city people?

You do not need a car as the train system and connecting trams and buses are excellent. You can buy an OV-Clipkaart for buses and trams that works everywhere.
Leiden is really beautifulp, a university town of 100,000 with flower decked canals.The Pilgrims lived here for ten years after they fled England and before they sailed off on the Mayflower for the New World.
We love The Hague. It is alao urban but easily walkable, royal palace, parliament, and the wonderful Mauritshaus Art Museum. It is small and filled with Dutch treasures. and you can take a tram out to the beach for lunch.Also, a day teip to Kindersijk is special. Train to Rotterdam, tram feom train station to waterfront, waterbus to Kinderdijk, a UNESCO site of 19 windmills , left where they worked to reclaim the land.

March isn't really the best time of the year to be outdoors, but you might be lucky...
In Amsterdam you'll like the Cromhouthuis: a wonderfully preserved canal house. And in addition to the Anne Frank Huis, you should visit the Portuguese Synagogue, part of the Jewish Historical Museum. The »hidden church« of Our Lord in the Attic might also appeal to you.
For your summer visit I recommend Enkhuizen, 1 hour from Amsterdam by direct train. A historical port city with a wonderful combination of former glory and village feel. In season there is also the open-air Zuiderzee Museum. Kinderdijk is also best kept for the summer when you have a direct river ferry link from Rotterdam. But out of season, when the weather is fine, you can also easily reach it with a bus from the railway station in Dordrecht, which is also a city worth a visit.
In Den Haag you'll like the Panorama Mesdag, one of the few panoramas left in the world.
Leiden and Delft are nice outings from Den Haag. The train will bring you to both within 15 mins. Delft is the prettier of the two and Leiden the livelier. In Delft you shouldn't miss the Old and New Churches. In Leiden you can visit the windmill »de Valk«: a 5-minute walk from the railway station (you can't miss it). I wouldn't look for »flower decked canals« in Leiden. In season some bridges have flower baskets hanging on the railings.
A car is indeed useless: there isn't much you can't reach by public transport. This transport planner gives you real-time information on all types of public transport.

Thanks, TonFromLeiden and Suki! I appreciate the suggestions of historic and/or lovely places to visit and especially the link to the transit planner. All this train travel will be an exciting new thing for my husband; he hasn't spent much time in Europe, and he's used to driving vacations. But I'll remind him it's all part of the experience!

Be careful about the Hague. I found it less appealing to walk around than other Dutch cities, and mainly of interest for its museums. Since these don't interest you, you will probably want to focus on day trips from there (in other words, you may spend a lot of your time in the Hague trying to get out of it!). Of course, if you can stay with friends, presumably for free, that's different. But if you have to pay money to stay there, you'd probably be happier in a smaller and less urban destination, which should also have the advantage of being cheaper.

Let your daughter arrange OV chipkaarts. Marvellous system which works nationwide on the bus, tram, metro, train and even the river ferry around Rotterdam.

I suggest that you think more about what you may mean by "not ... city people." To me, the Netherlands and Belgium are filled with banal postwar suburbs that have only modest charm. You are not going to be visiting many wooden villages where "Band of Brothers" was "filmed" and the people are wearing traditional costumes. What exactly are you hoping to "do" outside the cities - especially in winter? I like all the cities that have been mentioned so far. If the weather is good, De Hoge Veluwe National Park pairs well with the Appeldorn palace.

On our last stay in the Netherlands, I was sorry to not have time for the Delta Works. There are several important dance companies in the Netherlands. The food is better in Belgium. Alkmaar is touristy but may be what you meant by non-city. Without outdoor flowers, I would spend a morning at the weekday wholesale flower auction at Alsmeer. I suppose they bid on Spanish and African blooms at that time of year, as well as greenhouse flowers.

You already said you're not the big on art. But if I were in the Netherlands in March, I would go to Tefaf Maastricht
March 10-18, 2018. This is a really, really important retail art fair. I'm sure the city is jammed for it - you don't just go there on a whim.

The Netherlands and Belgium are filled with banal postwar suburbs
That's true, but between all those built-up areas there is a lot of beautiful countryside to be found where you forget you are in a heavily populated country.
A combination of train and bus brings one within one hour from Den Haag to historical Oudewater. From there you can have a 1,5 hour walk to the charming village of Linschoten along the twisting small river of the same name. A 10-minute bus drive brings you to Woerden to pick up the train.
A 20-minute train ride brings one from Amsterdam Centraal to Abcoude. A short walk from the station finds you at the river Gein, along which a quiet road leads to the systematically overlooked historical city of Weesp.
Take the bus from Alkmaar to Schermerhorn, with a beautiful windmill complex used for the reclamation of the Schermer polder. A country road, closed for traffic, offers a pleasant 1,5 hour walk to the village of De Rijp, once famous for its herring and whaling industry. A half-hourly bus gets you to the railway station in Purmerend.
It's all there if you know where to look

Interesting feat of hydraulic engineering and worth to visit is the Maeslant Barrier west of Rotterdam. Not so easy to get there with public transport, you have to take the train to Hoek van Holland and the final few km with a taxi. There is a visitor centre offering guided tours. http://www.keringhuis.nl/index.php?id=37
Having a car you can see more of the Delta Works, a huge system of dams and other structures protecting the southwestern corner of the Netherlands against the sea.

Thanks again for the continuing helpful suggestions! TonfromLeiden, that’s exactly the kind of local info I was hoping for - interesting small towns and landscapes that we can easily reach. Wil, I actually visited the Maeslant barrier a few years ago and got the tour; fascinating! Harold, thanks for your point about The Hague. Our hosts are either in Zoetermeer or Rijswijk, not right in the city - any preference between those two? And i’m hoping to catch some classical music concerts while we are there; possibly dance though that is less interesting especially to my husband. Planning continues...

Zoetermeer is a »new town«; Rijswijk a suburb. Pick the one which is the most practical for public transport.

Visiting the castle - Kasteel De Haar - is a very nice non big city outing. The castle is magnificent and the surrounding small (tiny) village of Haarzuilens is interesting to walk around. The kitchen, with all of it's brass pans, is wonderful to see. The gardens and the moat are so well maintained it is beautiful. The castle is not far from Utrecht (between Leiden and Utrecht). It is best to have a car to visit here. If you have a car, the village of Loenen aan de Vecht is a nice place to stop and walk around. (If you are in the area.)

You could go for a day trip and visit Kampen - beautiful old city and then head to Giethoorn and take a boat ride around the thatched roof village. Long drive to get there but beautiful homes to see.

De Haar is a late-nineteenth-century fantasy castle, but I second the suggestion of Kampen, one of a string of former Hansa cities along the IJssel river. Kampen has the best river front, and the most charming is Zutphen. All easily reached by train (Zwolle would be the best base). Even tiny village-like Hasselt (a short bus ride from Zwolle) prides itself to be a Hansa city. (Across the river you have the mid-14th-century polder Mastenbroek, one of the oldest land reclamations in the Netherlands.)
Just north you'll find the national park »Weerribben«, which should be good for walking, but I assume it will be difficult to reach by public transport.

Ton - Maybe it has to do with a certain type of “polder” (reclaimed land) that (some) regard Mastenbroek as one of the oldest land reclamations in the Netherlands, however land reclamation in the Delta Region on a massive scale dates back after the flood of 1134 A.D., some two centuries earlier as the reclamation of Mastenbroek.

You say that you are not city people. Well then, how about one of the islands... Texel etc.?

They will be lovely in the summer. It's where the locals spend their vacation.

In March, your degree of enjoyment would depend on how outdoorsy and weather-resistant you are. It's not going to be warm, but you would probably have the island pretty much to yourselves. If you happen to be into birding, there should be quite a bit of migration going on at that time of year.

The chain of islands continues on the German side, and German islanders say "There is no such thing as bad weather. There is only bad clothing." :-)