MOSCOW, March 31 (Reuters) - The Kremlin said on Friday that all accredited foreign journalists could continue to work in Russia, a day after a Wall Street Journal reporter was remanded in custody on espionage charges brought by the FSB security agency.

The Kremlin said Evan Gershkovich had been carrying out espionage "under the cover" of journalism. Russia has not published any evidence to support the charges - the first such case against an American reporter since the end of the Cold War - which have been denied by the WSJ.

"All journalists who have valid accreditation here - I mean foreign journalists - can and do continue their journalistic activity in the country. They do not face any restrictions and are working fine," Kremlin spokesman Dmitry Peskov told reporters on Friday.

Gershkovich, who is an accredited reporter, was arrested in Yekaterinburg on charges decried by the U.S. administration and those who know him as bogus. A Moscow court ordered him to be held in pre-trial custody until at least May 29.

The case - the most serious action against a Western reporter in Russia for decades - has threatened to further worsen already abysmal U.S.-Russia relations.

Civil rights groups criticised it as the latest escalation in Russia's campaign to stifle all independent journalism, which has intensified with the toughening of censorship laws since Russia sent its armed forces into Ukraine.

While a number of Russian journalists have been imprisoned during the two decades since Putin came to power, foreign reporters who have fallen foul of the authorities have generally simply been expelled.

Peskov on Friday repeated an assertion that Gershkovich had been "caught red-handed", but declined to elaborate on the details of the case, which have been kept secret even from the reporter's legal team.

He also urged Washington not to respond by expelling Russian journalists working in the United States.