Palat, 26, had 52 points (17 goals, 35 assists), five power-play goals, 15 power-play points and three game-winning goals in 75 games with the Lightning last season, when Tampa Bay did not qualify for the Stanley Cup Playoffs.
"I've been training in Ostrava (Czech Republic) for about a month and a half," Palat said. "I want to be 100 percent ready. [Last] season didn't meet neither our own nor our fans' expectations and we have to do a lot to make it up. And I believe we [will] succeed."
Selected by Tampa Bay in the seventh round (No. 208) of the 2011 NHL Draft, Palat has 218 points (74 goals, 144 assists) in 307 regular-season games and 29 points (14 goals, 15 assists) in 46 Stanley Cup Playoff games.
"I like that pressure," Palat said. "I'm one of the guys with the longest tenure [on] our team and it's natural that there will be a lot of expectations about my performance. And I'm ready to meet those expectations."
Tampa Bay this offseason traded its other restricted free agent forward, Jonathan Drouin, to the Montreal Canadiens for defenseman prospect Mikhail Sergachev, and Drouin signed a six-year, $33 million contract with Montreal.
Tampa Bay has signed free agent wing Chris Kunitz and defenseman Dan Girardi, and re-signed defenseman Andrej Sustr. The Lightning are about $3.2 million under the NHL salary cap of $75 million, according to CapFriendly.com.