Radisich showed early promise in single-seaters. In 1983 he was Formula Atlantic runner-up, earning the prestigious Driver to Europe award. He raced in British Formula 3 in 1985 and 1986 alongside Damon Hill, then competed in Indy Lights and Formula Super Vee with moderate success. He finished second in the 1990 Bathurst 1000, a result that pointed him firmly toward saloon car racing.
Radisich joined the British Touring Car Championship in 1993, driving a Ford Mondeo prepared by Andy Rouse Engineering. Despite competing in only half the season he finished third in the championship. He was back with Andy Rouse in 1994 and again finished third, this time behind Gabriele Tarquini and Alain Menu.
His most celebrated achievements in the BTCC era were back-to-back Touring Car World Cup wins at Monza in 1993 and at Donington Park in 1994, both driving the Rouse Mondeo. These victories against the field's fastest manufacturers' cars established him as one of the most respected touring car pilots of the period.
The latter years of his BTCC stint were less rewarding. By 1995 and 1996 the Mondeo's development had plateaued; when West Surrey Racing took over the Ford programme for 1996, Radisich managed no podiums and finished 13th overall. A new Mondeo in 1997 was similarly off the pace. A season with Peugeot in 1998 yielded little before he left Europe for Australian touring cars.
In 1999 Radisich joined Dick Johnson Racing, replacing John Bowe, and remained with the Queensland team through the end of 2002. He then moved to Briggs Motor Sport, which was later acquired by Triple Eight Race Engineering in 2004. Across his V8 Supercars career he took four race wins and eight additional podium finishes.
For 2005 and 2006 he signed with Team Kiwi Racing, driving a Holden for the first time. He helped TKR to their first ever podium finish at the Shanghai International Circuit, and ended 2005 fourteenth in the championship — the team's highest-ever season result.
The 2006 Supercheap Auto 1000 at Mount Panorama proved disastrous. While attempting to overtake Nathan Pretty in The Chase section at around 200 km/h, Radisich ran wide onto dirt, could not brake in time, and hit a tyre wall head-on, rolling the car onto its side. Rescue crews had to cut away the roof to extract him. He was flown to a Sydney hospital and was found to have broken an ankle and his sternum, ruling him out for the remainder of the season.
Radisich returned to TKR for 2007 on a new two-year deal, but Ford Performance Racing terminated its supply contract with the team in May 2007. Radisich exercised an exit clause in his own contract immediately, stating TKR was in breach and he was departing with immediate effect.
He subsequently joined the Toll HSV Dealer Team for the 2007 endurance rounds. Paired with Rick Kelly at Sandown, the duo finished second, 2.8 seconds behind Craig Lowndes and Jamie Whincup. At Bathurst, paired with Craig Baird, persistent braking problems and a strategic team decision to protect their championship contenders' machinery led to a retirement on lap 137 of 161.
Returning to the Toll HSV team for 2008, Radisich suffered a catastrophic accident during practice for the Bathurst 1000. His throttle jammed open at McPhillamy Park Corner and the car struck a gap in the tyre barriers at full speed. He sustained fractures to both ankles, reopened the sternum fracture from 2006, fractured lumbar and thoracic vertebrae, cracked ribs, and bruised his lungs. The injuries ended his professional racing career.
He made a brief return to competition in the 2018–19 BNT V8 Championship in New Zealand, racing a Ford Mustang S550 in an invitational non-championship class.
Radisich served as team manager of Super Black Racing until the team disbanded at the end of the 2016 season. He is managing director of Aegis Oil, a New Zealand oil company founded by his father Frank, and has worked as an in-studio co-host for the BTCC coverage on New Zealand's TV3.
Radisich's two Touring Car World Cup victories remain the pinnacle of his career and stand as the only back-to-back wins by a driver from outside the main BTCC manufacturers' ranks in that era. He was one of the few New Zealanders to achieve consistent success in European touring car racing during the highly competitive Super Touring era of the 1990s.