Laurie Anderson, Trisha Brown, Gordon Matta-Clark, Barbican Art Gallery

I can still remember the excitement of pounding the pavements of SoHo in the early 1970s. Nowadays, this part of downtown Manhattan is awash with expensive restaurants, boutiques and smart galleries, but then it was a scruffy industrial area of warehouses and sweatshops. The factories were closing and the container trucks leaving, though, and artists were gradually infiltrating and turning the huge empty spaces into studios where they often lived illicitly.

Sleeping on a platform in the workshop of an industrial designer on Broome Street, I felt the thrill of being in the right place at the right time; the air was positively buzzing with creative energy. The artists were more interested in generating ideas and creating interactions than in producing products. The city was on the verge of bankruptcy and they were keen to engage with the social and political situation; the work they made was not for sale to rich collectors, but was part of an ongoing dialogue about the surrounding dereliction.

Jenny Holzer fly-posted her Truisms onto grotty warehouse walls. Sayings such as "Private property created crime" and "Potential counts for nothing until it's realized" were like incendiary devices lobbed into the public consciousness. Others were keen to debunk the preciousness of art. Larry Poons tipped buckets of paint onto his canvases, Fred Sandback defined space with lengths of string, Richard Serra splattered molten lead along the join between wall and floor and Alan Saret created a knee-deep installation from the cotton waste left by the former factory owner.

Others were breaking down barriers between art forms. I wandered into the Kitchen, a venue that sprang up to show performance art and film, a medium artists were just beginning to explore, and discovered Laurie Anderson. I was utterly seduced by her electronically modified voice and ingenious gadgets, which included a violin with a tape head attached that could read the audiotape strung to her bow. Waving another bow back and forth across a projector beam, she "caught" the image with the moving wand. It was magical.

The memories are so vivid that an exhibition which tries to capture the vitality of the time will inevitably be disappointing. The problem is not just personal, though. The Barbican is presenting work that has influenced subsequent generations to such an extent that the attitudes and ideas it embodies are now commonplace; and its hard to convey just how radical they seemed 40 years ago.

The exhibition pays tribute to Anderson and two other downtown movers and shakers – the dancer Trisha Brown and Gordon Matta-Clark, an architecture graduate who was more interested in exploring the urban wasteland than in building gleaming skyscrapers. He gathered together like-minded friends under the name Anarchitecture that, in combining “anarchy” and “architecture”, encapsulates his combative relationship with the built environment.

He hired an industrial container, parked it on Greene Street and, using materials salvaged from the streets, divided the interior into small rooms. Friends were invited to stage a happening in and around the ad hoc spaces; it poured with rain so umbrellas featured large in the ensuing shenanigans, which were recorded as a film now being projected beside a mock-up of the container.

Open House (1972) is also recreated daily by five dancers, but they can’t hope to capture the laid-back anarchy of an event that was as much about claiming the street and establishing a social scene as staging a happening. Lacking any such raison d’être, it has become a performance without a purpose that inevitably feels aimless.

Matta-Clark photographed the interiors of abandoned buildings, then began cutting away sections of floor or wall to open up new vistas and create new spatial experiences. Preserved on film are more savage interventions such as Splitting (1974) (pictured above) in which he sliced in half a two-storey clapboard house in New Jersey before tipping one half off its foundations.

Photographs and films were useful ways of documenting such interventions, but the status of these records became a hotly debated issue. Were they art works in their own right? As though to ensure that something concrete remained, Matta-Clarke preserved the corners of the roof as free-standing sculptures which may have seemed a good idea at the time, but now feels like a cop-out.

Dance is by nature ephemeral; luckily many of Trisha Brown’s early pieces were recorded. She was keen to work with non-professionals on pieces created especially for urban spaces. It must have been thrilling to watch a man walking down the side of a seven-storey SoHo building with only a harness to prevent a fall, but we have only photographs to remind us. Fortunately Spanish Dance (1973) was recorded on film. The piece is extremely seductive in its simplicity. The dancers stand in front of each other; holding her arms above her head, the rear dancer sashays forwards until she reaches the next one; the pair then sashay in unison until they reach the third person who joins them, and so on, until the crocodile of swaying dancers reaches the far wall and stops.

Brown referred to simple moves like these as "pure movement" because they contain no symbolic, functional, emotive or associative elements. Three of her dances are being staged; Planes and Floor of the Forest are more interesting as ideas than actualities, but site-specific pieces like Walking on the Wall still have the power to interfere with one’s awareness of space. Suspended from harnesses, the performers walk horizontally along the wall, creating a dizzying sense of disorientation that fires the imagination.

The star of the show, though, is Laurie Anderson. She is such a superb storyteller that her work never loses its relevance and immediacy. There’s the story about the psychiatrist who hasn’t noticed the lip marks on her mirror. The narrator points them out and subsequently it transpires that the shrink’s daughter creeps in every day to kiss the mirror. “At that point”, Anderson concludes, “I realised that we were seeing things from such a different point of view that I would never have to see her again.”Rest your head on Talking Pillow (1977) (pictured left), listen to the subtle cadences of her voice and drift into dreamland. “In this dream I’m on a tightrope tipping back and forth and trying to keep my balance. And below me are all my relatives and if I fall I’ll crush them,” she intones. "This long thin line, the song line, the shout, this tightrope made of sound – the only thing that binds me to this turning world, made of my own blood... Remember me is all I ask and if remembrance be a task, Forget me..." But with her wonderful imagination firing on all synapses, Laurie Anderson is the one we are least likely to forget.

• Laurie Anderson, Trisha Brown, Gordon Matta-Clark at Barbican Art Gallery until 22 May