Picked out by the sharp light of an early summer's day, Villa La Rotonda looks unnaturally perfect, its shallow dome echoing the low hill on which it stands, an exercise in pure geometry, a cross on a square on a circle, milky pale as if carved out of ice-cream.

The house has stood since 1567. It sits just outside the small northern Italian town of Vicenza, in the Veneto. This is the region that surrounds Venice, its capital, forming a lush agricultural plain in the shadow of the Dolomites. From Venice, it stretches towards Verona and Milan in the east, and south towards Bologna.

Vicenza is a pleasant little town, but if it weren't for the works of Andrea Palladio, a miller's son from Padua who – inspired by the architecture of ancient Rome – created the Rotonda and scores of other buildings in the town and across the Veneto, it is doubtful many tourists would search it out. Palladio designed some of the most eerily perfect buildings ever constructed. For centuries architects, artists, poets, and wealthy tourists have come to revere them – and gone home to imitate. His influence survives not just in aristocratic party houses such as Lord Burlington's Chiswick House in London, or mansions like the White House, but as anyone who has ever struggled with a baby in a buggy up slippery steps to a dank portico knows, it's there in countless town halls, libraries and museums across the world too.

Villa La Rotonda is the most famous of his villas; the Maser, on the outskirts of the village of the same name, and where Palladio died in 1580, is perhaps even more beautiful, cut into a hillside with breathtaking views and interiors covered in ravishing frescoes by Veronese. Both are Unesco world heritage sites, but there are scores more scattered across the Veneto, and literally thousands more by other architects.

"Perhaps the art of architecture has never reached such a height," the German poet and philosopher Johann Wolfgang von Goethe wrote in his notebook, when he came to La Rotonda in 1786 on a personal pilgrimage. Goethe evidently managed to get inside, since he described the domed central hall as "of the most beautiful proportions".

Many pilgrims have been less fortunate. Most of the privately owned houses have never opened to the general public, and the ones that were open kept odd and unpredictable hours, with no central information point to help plan visits. But this spring, a new initiative launched to promote these treasures, with private, public and commercial owners co-operating for the first time. A Ville Venete website now details the project of "opening up" the houses, and so I managed to get into several Palladio villas across the region, some of which host guests overnight.

I strolled at dusk through a private wood scattered with carved stones from older buildings right back to the Romans; tried not to stare too hard at sideboards covered in photographs of half the titled families in Europe that I recalled vaguely from Hello! royal wedding spreads; stood in frescoed halls just as the Venetian aristocrats once stood gazing out at their vines and fields stretching to the horizon.

I even stayed in a huge frescoed room in another palace, the Villa Cornèr della Regina, near Treviso – a charming town bypassed by millions of tourists every year on their way to Venice – and close to another Palladian beauty, the Villa Emo.

The Villa Cornèr della Regina was built for a nobleman's daughter who became the queen of Cyprus. After an unhappy period as a timeshare, it is now a hotel owned by an Australian-born Italian who is big in LED lighting: my ceiling was a 17th-century gem, the blue glowing panel by my bed a bewildering puzzle in turning out the lights without calling for room service. It still feels more like the wedding-and-events venue that it has been operating as, rather than a hotel, but anyone who gets a room in the main building can live like a Venetian count.

At La Rotonda, as we came out from the cool shadowy interior – where each huge window frames a different landscape like a painting – we found a disconsolate Frenchman under the loggia trying to peer in through net curtains.

He couldn't have known that the man in the pistachio-green cashmere sweater leading our little group was the Count Niccolo Valmarana, descendant of one of the oldest families in Venice, whose grandfather sold thousands of acres of farmland a century ago to buy and restore the derelict masterpiece. He did see the bunch of keys and hared across to plead to step inside, even for a moment. Impossible, said the count, he should come back the following day when the house was open. Impossible, said the Frenchman, he would be 100 miles away by then. Regrettable, but impossible, said the count, implacable.

The Frenchman cast me a baleful look as I followed the count out into the sunshine. If only he'd known to check the opening time on the Ville Venete website. The website, like the project, is a work in progress – but there are maps giving information about the villas, making it possible to see how excursions could be planned from the famous but tourist-choked cities of Verona, Padua and Venice to groups of villas in countryside where tourism has barely begun to make an impact.

Some, as with La Rotonda, are perfectly accessible by public transport, others could happily be toured by car, from a base in one of the towns or even by staying in one of the villas now offering self-catering or B&B. The initiative is driven by a noble desire to share their treasures and their history, as many of the owners insist, but also, as some admit, by the pragmatic reality of the state of the Italian economy and new property taxes. The householders are facing new taxes by the square metre – "as if it were a flat in Milan!" one snorted in outrage – which is a big headache for anyone living in a house with a facade the length of a village street.

Daytrippers will find the experience different from visiting National Trust properties and many stately homes in the UK, which are increasingly displaying themselves from basement pantry to servants' attic bedrooms. Generally only the grandeur of the entrance halls and a suite of magnificently decorated but surprisingly modestly-scaled family rooms around the edge will be on display. However astonishing the facades look, flanked by fountains, bristling with statues of gods, most of the houses are only one-room deep, and were built with a practical function: they were summer palaces, refuges from the smells and the fevers of Venice's hot season, but they were also farming estates. Without the Veneto, Venice would often have gone hungry.

The visitor is far more likely to meet some family member, conducting a tour or offering a glass of prosecco from the estate, than say, the Duke of Devonshire at Chatsworth. The experience can be slightly surreal. Over dinner in the grandeur of the frescoed hall of the Villa Ca' Marcello, near Treviso, Count Alberto Passi de Preposulo and Count Leopoldo Marcello sat playing a sort of Veneto top trumps, trading knights and dukes, judges and princes. Both had handsome sons working in public relations for Giorgio Armani.

Count Leopoldo's ballroom floor was recently cracked by an earthquake, Count Alberto merely had a ceiling fall down– "paf! Just like that!". Count Alberto sells produce from his own vines, such as cherries, apricots, apples, pears and quinces, whereas Count Leopoldo said: "I produce nothing, agriculture is finished." We eavesdroppers declared the contest a draw.

Travelling by motorways and main roads, the flat landscape of the Veneto can be a dismal procession of half-abandoned industrial estates, suburbanised villages, tin sheds selling furniture or plumbing supplies, and a startling number of McDonald's branches. However, the villas, with fields of grapes or vegetables coming almost up to their walls – the Ca' Marcello is unusual in having that romantic woodland garden, for most the soil was too rich to waste much land on pleasure gardens – stand in a landscape that has hardly changed.

There are currently no visitor facilities except a small shop at La Rotonda, but Niccolo revealed he is planning to open a cafe and bar in the farm buildings next year. This was news to his neighbour in the villa across the lane, who just happens to be his cousin, Countess Carolina Valmarana.

Last year she opened her own charming bar – open every day even when the house is closed – in the coach house of her beautiful Villa Valmarana ai Nani (Stradella dei Nani 8, Vicenza, villavalmarana.com), whose principal rooms are completely covered in frescoes by two of the most famous Venetian artists, Giambattista Tiepolo and his son Giandomenico.

Count Alberto Passi de Preposulo, a campaigner for organic produce, slow food and eco-tourism, is also the president and powerhouse of the Ville Venete project, driving the others forward by example. His wife helps make the jams and preserves which are sold in their farm shop. He leads tours of his beautiful Villa Tiepolo Passi, including the endless suite of rooms through which he and his four brothers used to race on their bicycles.

He also researches ancient recipes, shares a glass of wine with the lodgers, speaks at travel trade shows, and hires local musicians and poets to entertain at the dinners he has launched in one of the estate buildings.

So far, just 50 of the villa owners have joined him, leaving some 4,188 to win over. He is not a man to be daunted by such a statistic. "There is far to go, but it is well begun."

• The trip was provided by Veneto Promozione. More information on the villas of Veneto can be found at villevenete.net and veneto.to. Villa Cornèr della Regina (+39 0423 481440, villacorner.it) offers junior suites on a B&B basis including a welcome drink from €139 per night. British Airways (0844 493 0787, ba.com) has flights to Verona from Gatwick from £79 return. Ryanair has flights to Venice Treviso from East Midlands from £29