We are looking for a multi-day trek in Tanzania: we are less interested in scaling peaks but very interested in being part of nature and seeing villages. What would you recommend?

I think you'll find the Usambara Mountains in northern Tanzania pretty much perfect for the type of hiking you're after (another possibility, all be it one just over the border in Kenya, is the Loita Hills).

There are all kinds of hiking possibilities in the Usambara Mountains from half a day to a week or more. There's nothing technical or difficult about the walking (though make sure you have plenty of wet weather gear!)

The main launch point for treks is the small town of Lushoto. If you're not booking an organised trek and don't mind doing things DIY you should be able to find a guide here, although there's no official registered guiding agency. A good bet can be to ask at guesthouses for help finding a good guide. You wouldn't likely be a part of a set group as the guide would walk with just your party.

You won't need tents as its more village to village trekking with simple guesthouse/homestay style accommodation in each village. You also won't need to carry anything more than snacks as food is always available in guesthouses. It's all quite organised considering the small number of people who hike here.

As for routes your guide will likely have suggestions, but Wikiloc also has a few user recorded routes which could serve as a good starting point. It's a very beautiful area. With big escarpments, drop-offs and varied countryside and people will try hard to help.