My wife, Maggie, and I are accidental tourists in the most literal sense. Serendipity dictates our itinerary. We try to keep an open mind and trust the gorgeous prospect of chance.

The word serendipity was coined by the 18th-century English aesthete Horace Walpole to describe finding something of value while looking for something else. It derives from the Persian fairy tale "The Three Princes of Serendip," whose roving heroes, Walpole wrote, "were always making discoveries, by accidents and sagacity, of things they were not in quest of."

Serendipity allows a fresh response to place that would be blunted by a package tour: We find it hard to enjoy the moment if we have to worry about where we're headed next. We'd rather improvise, even if that means missing something extraordinary. Our travel strategy is not unlike the one adopted by the humorist S.J. Perelman, who deliberately went in search of "arduous situations" on the road.

Last summer, our 36th together, we opted for Greece. In the past, we'd been enchanted by its islands Crete, Santorini, and Samos. We loved their loud, welcoming people, their ruins, their seafood and beaches, and their peaceful, easygoing pace. If not for the mandatory smoking, and the waiters who decide to retire after seating you but before taking your order, they would be our ideal.

Our original plan, vague as it sounds, was to fly to Istanbul and then take a ferry to Rhodes, home of the Colossus and many romantic myths. A friend with a map of Asia Minor assured us that the island was just off the Turkish coast.

OK, so we never actually consulted the map. How hard could it be to hop from one place to another? (Then again, Odysseus, Greece's most serendipitous traveler, took 10 years to get from Troy to Ithaca.)

Not that we travel totally unprepared. Before we leave, we buy guides - though we give them only the most cursory of reads.

I concede that without prior research, a tourist might miss the boat - quite literally in our case, since there is no ferry from Istanbul directly to Rhodes. Instead, we took a connecting flight through Athens, which saved us a 12-hour bus ride: We're accidental tourists, not irrational ones.

Everything good

We boarded a bus to Kolona harbor, where the 12-story Colossus loomed over the city of Rhodes before being toppled, in 226 B.C., by an earthquake that also leveled the city. "Even lying on the ground, it is a marvel," wrote Pliny the Elder, so huge that "few people can get their arms around its thumb."

Following the principle of sinistrality, we made a series of left turns and arrived at a vast Crusader fortress, its menacing towers and swallowtail battlements stained honey and ocher by the sun. Alas, the restaurant recommended by a fellow bus passenger was closed.

Standing by the door was a Rhodian with an eggplant-shaped nose that Pliny the Elder might have had trouble wrapping his arms around. Sensing our disappointment, he motioned for us to follow him. We plowed under flying archways and down narrow alleys where tourists were thicker than bees. Finally, we arrived at a taverna that wasn't in any of our guidebooks. We were the only customers.

We dined on tiny shrimp lightly fried in their shells while our accidental guide expounded on the Rhodian love of life and chaos. His own life, he said, had one ruling principle: Everything good is discontinued.

Three moments

He advised us to take a bus to Lindos, a seaside resort on the island where his mother had a house to rent. We did, though Lindos poses a serious challenge to the dedicated serendipper.

A half-century ago, Perelman rhapsodized about the village's "steep huddle of whitewashed dwellings nestled below its acropolis" and "shelving beach encircled by jagged hills and improbable blue water." Since then, Lindos has become thoroughly anglicized. Every sign is in English, everyone speaks it, and the bars and bistros go by twee names like the Sunburnt Arms and the (mercifully shuttered) Lindian Indian.

The winding, cobblestone main drag is made virtually impassable by masses of combative shoppers. They forage through tables and bins heaped with curios, T-shirts, and calendars featuring the "rebel street cats of Lindos."

Fortunately, cars are banned, and the town lends itself to walking. On the way to our rental house, we stopped and lounged under a lime tree by a wall. The late-afternoon sun turned the wall from white to yellow to green to blue. Such small, still moments are what serendipitous travel is all about.

We experienced two more transcendent moments in Lindos. The first came while we were climbing to the Temple of Athena, a palimpsest of ruins overlooking the Aegean. The only sounds were the rustling wind and the distant meowing of rebel street cats.

The second moment unfolded the day before our departure. On a lark, we strolled down to Perelman's "shelving beach" for a dip in St. Paul's Bay, a shallow cove ringed with mountain goats where the apostle is said to have put ashore on his way to Rome.

On this afternoon, the heavens were filled with sculptural clouds, the sea was clear and pleasantly warm, and all was, well, so very Greek. As night fell, we were still swimming. We turned our heads to breathe and see a last lemony fragment of sky. An instant later, it was gone. As any Walpolian will tell you, everything good is discontinued.

Franz Lidz is the author of "Unstrung Heroes."