* Rate cut seen in May or June

* March inflation figures due on April 12

* Policy decision on May 8

(Adds Thedeen comments on demand and the economy, rate cutting plans and the Swedish crown in paragraphs 7-11 and 14)

STOCKHOLM, April 8 (Reuters) - The inflation outlook has brightened and Sweden's central bank should be able to start cutting rates soon, but there are still risks of setbacks, Riksbank Governor Erik Thedeen said on Monday.

At its most recent meeting on March 27, Riksbank kept its key interest rate unchanged at 4.00% but said there was a strong chance it could start cutting rates next month.

However, rate-setters have pointed to a weaker Swedish crown and the chance that the promise of rate cuts could lead to a consumer boom and higher prices as possible obstacles to easier monetary policy.

"Exchange rate developments in particular create uncertainty," Thedeen said in the summary of a speech published on the central bank's website. "A weaker krona could provide unwanted inflationary impulses."

Sweden's economy slowed sharply in 2023 but the downturn looks to have bottomed out.

Lower rates will help households which have cut back on consumption as they face higher mortgage payments.

But Thedeen said the Riksbank was wary of stimulating demand too much if it promised rapid policy easing.

"We actually think the economy actually could grow a little bit stronger, so all growth is not of course negative news," he said.

"But if that comes too quickly so it affects price-setting behavior, that of course would be unwelcome."

Thedeen said the European Central Bank and the U.S. Federal Reserve's rate-cutting plans were important for the overall picture, but it would be the outlook for inflation in Sweden that set monetary policy.

"We don't condition our decision on specific actions by specific central banks at specific dates," he said.

The ECB is seen cutting rates in June, but with the U.S. economy stronger than expected, cuts by the Fed could be delayed for longer than markets are predicting, potentially weakening the crown.

With the outlook uncertain, Thedeen said the Riksbank is likely to cut rates cautiously when it gets started.

"I think that we underestimated the effect of the krona on inflation and we don't want to do that mistake again," he said.

The Riksbank's rate path points to three rate cuts in total this year from six remaining policy meetings.

Inflation figures for March are due on April 12 and the central bank's next policy decision will be announced on May 8.

(Reporting by Louise Breusch Rasmussen, Simon Johnson; Editing by Terje Solsvik, Stine Jacobsen and Emelia Sithole-Matarise)