LA PIAZZA in Goodramgate is a friendly Italian family-run restaurant. It is also a great place to have a coffee, says chartered accountant Philip Thake, and they do a mean lemon cake.

Today, however, our attention isn’t on the smells drifting out of the kitchen. Our gaze is fixed firmly on the ceiling.

This is a common reaction, says Giuseppina Comito. As the wife of the restaurant’s chef, Mario, she has seen plenty of visitors come in and stare at the ceiling. “They can’t take their eyes off it, sometimes,” she says.

It is easy to see why. The ceiling soars. Great oak timbers reach up to a dizzy height, interlacing in a way that suggests the inside of a church. You’d think you were in a great medieval hall, rather than a modern high street restaurant. In a way, you are.

Number 45 Goodramgate is a late-medieval half-timbered building that dates back to the late 15th or early 16th century: early Tudor, in other words. It would probably have been the home of a wealthy York citizen, with a shop in front, says Philip. And this splendid, high-ceilinged room would have been the dining hall.

The charms of 45 Goodramgate don’t end with this room. There is another seating area upstairs, reached by a twisty timber staircase. Like the dining hall, it is full of old timber beams, and there is a lovely deep-set window with what was possibly once a window seat looking out over Goodramgate. Philip pokes his head out of it for the benefit of Press cameraman Mike Tipping down below.

Philip, as well as being a chartered accountant, is chief executive of the York Conservation Trust. A registered charity, the trust was founded by John Bowes Morrell and his brother, Cuthbert, in 1945 as the Ings Property Company Ltd with the aim of buying, restoring and rehabilitating medieval buildings in the city. It now owns 91 properties, all told.

Many are hidden gems; the crowds hurrying past are possibly entirely unaware of the magnificent architecture behind and above the shop fronts.

The Conservation Trust wants to change that. Which was why, earlier this year, it published a walking guide to its historic properties. “We wanted to make people aware of what is here,” Philip says.

To get around all 90-odd properties on foot would be a walk of more than six miles. Philip estimates it would take three hours, even without spending much time looking at the buildings.

But if you want to know more about the hidden treasures lurking off every street in this beautiful city, it is well worth dipping into, and perhaps tackling a stretch at a time. Which is what I’m doing today.

Not all the buildings are open to the public. But many, in line with trust policy, are used by local businesses – shops and restaurants – so that visitors can at least go inside for a meal or to shop. Upstairs, many have been converted into flats.

The key to appreciating the overlooked glories of York is to look up as you walk about and see beyond the shop-fronts, Philip says.

Goodramgate is as good a starting point for a walking tour as anywhere. From La Piazza, it is no distance to Wealden Hall, at 51 Goodramgate – “one of the most important heritage buildings in the city” according to the Trust’s walking guide.

From the street, above the shop-fronts, the building is a magnificent black and white half-timbered building that juts out in layers in true medieval fashion. Upstairs, the front of the building is a flat, and at the moment, the downstairs shop is empty, so you can’t go in. The trust has a tenant who should be moving in soon, however, so hopefully that won’t last long.

Philip takes us in for a look around, stooping to pick up the post on the way.

The most remarkable feature of this building, he says, is the main hall. It soars over our heads, rising to the roof three stories above, and is another construction of wonderful oak timbers offset against white plaster.

The building is about the same date as No 45, and may have been built by a merchant from Kent, Philip says. Wealden halls of this type are generally found there. Down the centuries, the ancient timbers which line the inside walls and roof have twisted and shifted, leaving beautiful flowing patterns in the woodwork.

My favourite part of the building, however, is the narrow twisting staircase at the back, that leads up through the gloom to a tiny balcony and attic on the third floor, with a window peering out over the plots and gardens behind Goodramgate.

The trust owns another property in Goodramgate – No 60, at the end of Lady Row. But this time we give it a miss, and instead turn back and cut through Aldwark to Peasholme Green and St Anthony’s Hall.

Built in the mid 1500s by some of the richest merchants in the city, this was originally one of York’s four medieval guildhalls – the others are the Merchant Taylor’s Hall, the Merchant Adventurer’s Hall, and the Guildhall itself. St Anthony’s Hall was home to the Guild of St Martin; but being built on the site of a former chapel to St Anthony kept his name.

Today, it is home to the Quilt Museum, but it has gone through various incarnations down the centuries: a workhouse, a “house of correction” for minor criminals; a knitting school for poor children and, during the Civil War, an ammunition storeroom, military hospital and prison. Most recently, it was home to the Borthwick Institute.

The quilt museum is well worth a visit in itself. If you do go, make sure to look up at the ceiling of the main exhibition space on the first floor.

It’s another of those glorious, half-timbered black-and-white affairs, but much bigger than the others and resembles the timber hull of a great, upended ship. The roof is studded with carved wooden bosses, white roses presumably symbolising York among them, and also an intricately carved pig, associated with St Anthony, possibly because the Egyptian saint reputedly once worked as a swineherd.

Philip has another snippet of information. “In Tudor times they used to have archery practice in here,” he says, looking down the hall’s impressive length.

The trust bought the hall in 2006 and spent a small fortune underpinning it to preserve it for future generations.

“The building was twisting and leaning,” Philip says. They also did up the garden, creating a wonderful public space in the shadow of the city walls that is open to anybody to sit or wander or have their lunch.

Our final destination is Fossgate. The trust owns two properties here – No 13, home to Masons restaurant, and No 15/16, where Zik Zak is based. Sandwiched between the two is an unprepossessing green door with the words Morrell Yard on it. Behind this is a long, dark alleyway which leads ultimately to a lovely courtyard.

There were a number of cottages behind here which were, Philip says, in very poor condition. They were demolished and rebuilt, using the original bricks, and a further extension was added at the back, so that there are now ten homes clustered here.

What is most noticeable, however, is the round brick well near the yard’s entrance. It dates back to medieval times, and may be 17 metres deep. There could be some wonderful medieval artefacts down there, Philip believes: though if there are, they will be buried under more than 50 feet of accumulated muck, earth and rubbish.

Still, he dreams of doing an excavation one day. “It would be wonderful to have Time Team here.”

Morrell Yard is private, so if you do push through to have a look, make sure you respect the fact that people live here, Philip says.

• To find out more about the York Conservation Trust’s buildings in the city, visit yorkconservationtrust.org

The trust’s historical properties walking guide is available, priced £9.99, from the trust itself at 54 Bootham, or from WH Smith, Waterstones, Visit York or Fairfax House.