He moved to North America in 2009, spending one season with the Everett Silvertips of the major junior Western Hockey League before the Tampa Bay Lightning selected him 66th overall in the 2010 NHL entry draft.

After going un-drafted in the 2009 NHL Entry Draft, Gudas was invited to the Los Angeles Kings training camp that year but was not offered a contract.

[7] Gudas helped lead the Admiral blue-line, along with fellow Lightning prospect Mark Barberio, to the 2012 Calder Cup championship in a four-game sweep of the Toronto Marlies.

[14] Gudas suffered a MCL injury during the Eastern Conference Championship series, and missed the first four games of the Calder Cup finals.

[20] Gudas also received a match penalty against the Dallas Stars on 5 March 2014, although Tampa Bay general manager Steve Yzerman announced there would not be any additional discipline from the league.

[24] Gudas was suspended for three games on 2 December 2015, for an illegal check to the head of Mika Zibanejad during a 4–2 win against the Ottawa Senators the night before.

[31] On 6 January 2014, Radko Gudas was named to the Czech Republic men's national ice hockey team with teammate Ondřej Palát for the 2014 Winter Olympics in Sochi, Russia.

NBC Sports writer James O’Brien noted Gudas’ growing list of indiscretions in late 2016, writing that “If the Department of Player Safety had a speed dial list of regulars, Gudas would have a prominent spot on it.”[35] Hockey writers have described him as reckless[36] and as a player who consistently demonstrates no regard for opponents in vulnerable positions.

[39] On 10 October 2016, Gudas was issued a six-game suspension for an injury-causing hit delivered to the head of Boston Bruins' forward Austin Czarnik in a pre-season exhibition game.

[40] Gudas has also been involved in numerous questionable plays in which he has avoided disciplinary action despite placing opponents in danger or causing injury.

[35] This history of delivering dangerous hits while typically avoiding suspension has been referred to facetiously as Gudas' "signature skill" as a professional hockey player.

[41] In mid-February 2016, Gudas was ejected from a game for the third time in sixteen days after he injured New Jersey Devils’ forward Bobby Farnham with a late hit which targeted the head of a player not involved in the play.

[43] On 26 October 2017, Gudas was again ejected, receiving a five-minute major and a game misconduct after targeting the head of Ottawa Senators' defenseman Chris Wideman.

[44] Flyers' coach Dave Hakstol and GM Ron Hextall were reported to have met with Gudas in early 2016 to discuss his on-ice actions.

[45] Upon returning from this ten game suspension, Gudas immediately found himself in the news again after leaving his feet to deliver a hit to New Jersey Devils' forward Kyle Palmieri on 1 February 2018.

Though he avoided another suspension and claimed the incident was accidental, Palmieri said he believed Gudas no longer deserved the benefit of the doubt in light of his history, adding that he felt the Flyers' player had "run out of second chances.