Rađa began his basketball life in his native town, as a junior at KK Dalvin. From there he went on to KK Split to polish his professional career. He marvelled with Jugoplastika, as he and his friend Toni Kukoč led the team to the pinnacle of European club basketball twice in a row (1989 and 1990).
In late June 1989, the 22-year-old center got drafted by Boston Celtics in second round as the 40th pick. Right away, he publicly expressed willingness to go to Boston immediately "if the financial offer is good" thus joining fellow Yugoslavs Vlade Divac, Drazen Petrovic, and Zarko Paspalj who were on their way to the NBA that summer. However, led by general-manager Josip Bilić, Jugoplastika was adamant Rađa would not be released since they had him under contract until 1992. The entire case quickly turned into a months-long saga that played out in the Yugoslav media. Club's head coach Maljković even publicly called on Yugoslav Basketball Association (KSJ) to adopt safeguard policies, preventing players younger than 26 from transferring to NBA teams. After weeks of wrangling over his status, Rađa tried to force Jugoplastika's hand by physically going over to the U.S. in early August 1989 and signing a one-year contract with the Celtics, reportedly in the neighbourhood of $500,000. He also started practicing with the team at the Brandeis University facilities. However, seeing the situation as a clear case of contract poaching by Boston and its GM Jan Volk (who claimed Radja's contract with Jugoplastika is amateur and thus non-binding), the Split club wouldn't give up the legal fight, taking the case to US District Court for the District of Massachusetts, seeking an injunction to prevent Radja from playing for the Celtics on the grounds that he has a valid and legally binding contract with Jugoplastika. Following a hearing on 26 September, judge Douglas Woodlock ruled in their favour two days later thus preventing Radja from playing for the Celtics. Since Radja was physically already in Boston, bringing him back required some kind of an agreement. By mid November 1989 Jugoplastika and Celtics agreed terms under which the player went back to Yugoslavia to complete the 1989–90 season in Split before having the rights to his services transferred to Celtics effective June 1, 1990. Celtics paid an undisclosed sum to Jugoplastika that in turn agreed to let Radja go two years short of his contract completion.
Rađa celebrating with teammate Toni Kukoč after beating FC Barcelona at the European Champions Cup Final Four final game in Zaragoza in April 1990 to win Split's second consecutive title.
Radja was thus back in Split for Jugoplastika's 1989–90 season. Jugoplastika again won the Yugoslav League, its third consecutive domestic league title, as well as its second straight European Cup.
Despite the team's unprecedented success, as previously agreed, Rađa would not stay in Split past June 1990 thus relinquishing the chance to go for the historic European Cup three-peat (which the club, led by Kukoč, achieved the following year), but would not go to Boston either.
Instead of going to the NBA, Rađa went to Italy in August 1990, signing for money-rich Virtus Roma despite claiming all along that he wanted to join the Celtics. He changed his mind once the offer from Virtus, an ambitious and financially stable club bankrolled by the riches of the Gruppo Ferruzzi food company, came in, reportedly paying him in the $15–18 million range for a 5-year contract. Boston Celtics did not insist on him honouring his contract with them, instead agreeing to let Radja go to Virtus in return for an undisclosed amount, but retaining his NBA rights. Part of the reason Boston did not put up much of a fight when Radja suddenly decided to sign with Virtus was the July 1990 court decision against them following the motion by Radja's American agent Marc Fleisher after the Virtus offer came in. Taking advantage of an administrative loophole, Fleisher claimed that Radja's contract with the Celtics violated a provision of the agreement between the league and the NBA players that says, among other things, that one-year contracts cannot be extended. Special officer of the court heard the case and ruled in Radja's favour, against the Celtics.
In 2005, commenting on his summer 1990 decision to remain in Europe even after only a year earlier he seemed desperately intent on playing in the NBA, Radja said:
I was playing well. I was making a great salary in Europe. The thing about playing in the NBA was that there were so many unknowns. The NBA was more physical, because the players were bigger and stronger than in Europe. I also would have had to get used to an entirely different culture.
Rađa averaged 18.1 points in his first season with Il Messaggero (Virtus Roma enjoyed sponsorship from that popular Roman newspaper at the time). European sports journalists voted him the second best European player that season, shadowed only by his former teammate and friend Kukoč. He improved his scoring average each of the next two seasons with the Roman club, averaging 20.2 and 21.5 points, respectively. In 1992 he led Virtus to a Korać Cup title.
Rađa finally joined the Celtics in summer 1993, signing a three-year contract on July 9, four years after initial interest from both parties and the voided contract in 1989. Some two and a half weeks later, the team went through a shocking incident when the Celtics' leading scorer small forward Reggie Lewis passed away on the basketball court at the team's Brandeis University practice facility, suffering sudden cardiac death from a heart defect.
Playing alongside Dee Brown, 40-year-old veteran Robert Parish, and Rick Fox, 26-year-old Rađa averaged 15.1 points and 7.2 rebounds in his debut season during which he made $1.5 million in salary. With 32-50 record, Celtics missed the playoffs, finishing 10th overall in the East. At the end of the season Rađa was voted to All-NBA Rookie Second Team, along with Kukoč, at that time a rookie with the Chicago Bulls.
In early November 1994, at the start of his second season with the Celtics, Radja was looking for a contract extension on his existing three-year deal expiring in summer 1996. With his agent Mark Fleisher engaged in long negotiations with the Celtics brass led by GM Jan Volk, the deal was reached to add three more years to Radja's existing contract beginning with the 1996–97 season.
The 1996–97 season, Radja's fourth in Boston, was marked by a left knee injury forcing him to miss 57 regular season games. In January 1997 he underwent arthroscopic surgery on his knee. Celtics finished the season dead-last in their division with 15-67 record, the worst in the eastern conference, forcing a coaching change at the end of the season with Rick Pitino replacing M. L. Carr.
In June 1997 a trade that was to send 30-year-old Radja to the Philadelphia 76ers for Clarence Weatherspoon and Michael Cage fell through when Radja failed his physical with the 76ers on June 24, 1997. Apparently, the 76ers staffers examining Radja determined he had no cartilage in his left knee and estimated that "because his left knee is bone-on-bone, he can play games, but he can't practice afterwards because his knee will swell" and that "he can't play four games in six days". Celtics initially challenged the 76ers' decision to void the trade, but quickly dropped their arbitration request. Since Radja had three more years left on his guaranteed contract, if he could not pass another team's physical it meant that the Celtics would have to pay his entire remaining salary. Celtics were not convinced he could so over the coming weeks, the team and the player agreed a buyout of the three years that remained on his contract and then, on July 16, Celtics waived Radja.
In 2005, Radja talked about his exit from Boston:
I went to Pitino and asked him if I fit into his plans. With a new coach, I obviously wanted to know what he thought of my game. I loved playing for Boston and just wanted to find out if there was any possibility I might be traded, because I had heard some rumours. Pitino looked me right in the eyes and said, 'Dino, don't worry. You're going to be a big part of our offense. When we run a set play, the ball is going to go through you.' I left the meeting feeling great. Five days later, I found out I was being traded to Philadelphia. I can't tell you how much I felt betrayed. Either Pitino lied or something changed in a matter of a few days.
Over the course of his four seasons with the Celtics, Radja averaged 16.7 points and 8.4 rebounds in the regular season. In the NBA playoffs where he only made a single appearance with four games played, he averaged 15.0 points and 7.0 rebounds.
In the wake of his failed physical in Philadelphia and Rick Pitino's unwillingness to keep him on the Celtics' roster, Radja returned to Europe in July 1997, joining Panathinaikos, a rich and ambitious club bankrolled by the Giannakopoulos brothers who made their money in the pharmaceutical industry.
Finishing the previous 1996–97 season without trophies (after finally winning both the Euroleague and the Greek Cup in 1995–96 after years of trying) was deemed unacceptable and brought about big changes in the team. Head coach Bozidar Maljkovic (Radja's mentor from Split days) was already released during the season while his interim replacement Myhails Kokalis was let go in summer 1997 to make way for Slobodan Subotic. Also coming in alongside Radja and coach Subotic was 36-year-old NBA veteran Byron Scott from LA Lakers.
Radja spent two years in Athens, winning two Greek Championships, but returned in 1999 to his native Croatia to play for Zadar. He left Panathinaikos because of an encounter with the club president's son in the locker room after the game. The president's son allegedly cursed at Dino, but at that time Dino didn't know that the person he argued with was the son of the club president.
In 2000 he returned to Greece, joining Panathinaikos' long-time rivals, Olympiacos, in an unsuccessful attempt to regain the Greek Championship. There, on 16 October 2000 he scored first points for newly established Euroleague competition in a match against Real Madrid.
He returned, once again, to Croatia, joining Cibona for the 2001–2002 season. Radja finished his career in 2003, winning the Croatian championship with his first team, Split.