By Denis Pinchuk

ST PETERSBURG, Russia (Reuters) - Thousands of people protested against rising food prices across Russia on Thursday, highlighting one of the biggest challenges that will face Dmitry Medvedev when he takes over as president next week.

The cost of everything from bread to fuel and apartments is soaring as record prices for oil -- Russia's biggest export -- flood the country with petro-dollars.

Some analysts have warned that popular discontent over accelerating inflation could quickly spiral into a major political problem for Medvedev, who will be sworn in on May 7 to replace his mentor Vladimir Putin.

Around one and a half thousand people gathered on Nevsky Prospect, the main drag of Russia's second city St Petersburg, shouting "No to high prices!" and "Putin's plan means high prices".

"The rise in prices comes from our government's massive spending ... We have started to pay an inflation 'tax'," Putin's former economic advisor Andrei Illarionov told Reuters at the protest in St Petersburg, which is also Putin's hometown.

On the Pacific island of Sakhalin, whose southern tip is a short boat trip from Japan, 3,500 people demanded the government intervene to keep food and fuel prices down.

The islanders gathered in the main city Yuzhno-Sakhalinsk also asked for a raise in pensions and public-sector wages, said Svetlana Klyuzhina, a representative for the Federation of Trade Unions in Russia's Far East region.

The protests were part of countrywide marches to mark Labour Day, an annual public holiday that traditionally attracts large crowds demanding better treatment for workers.

SCEPTICAL

Price rises are the main headache for the government as the economy booms for a 10th straight year.

Government figures put inflation at 6 percent in the year to April 21, setting the government a difficult task if it is to meet its target of 9.5 percent inflation for the year,

Many people are sceptical of official forecasts and say prices are rising much faster than the government admits.

For decades under Communist rule prices for nearly all goods and services were set by the state. Today's government says it is committed to the market economy but is coming under increasing pressure to reinstate price controls.

In some regions officials have agreed with producers and suppliers to cap prices on certain products.

In Moscow, thousands of trade union members held flowers and waved blue and white flags as they marched through the city centre under sunny skies.

"We have low stipends and we have nothing to eat. We are all for freezing prices on food products," said Maria, a student at a local university who declined to give her last name.

"I am here so that the trade union will take care of us and to ask for our salaries to go up and become sufficient," said Raisa Vladimirovna, a Moscow university professor who gave only her first name and patronymic.

Protests against high food prices were also held in dozens of towns across Russia, the far eastern port of Vladivostok, the Caucasus city of Vladikavkaz and Stavropol in southern Russia.

Earlier this week, train drivers at two depots near Moscow went on strike over wages, causing disruption to commuter services. They threatened further industrial action. Strikes by public sector workers are extremely rare in Russia.

(Reporting by Denis Pinchuk in St Petersburg, Alexei Dovbysh in Vladivostok and Mikhail Antonov in Moscow; Writing by Amie Ferris-Rotman, edited by Richard Meares)