Voting booths were still being dismantled Tuesday night when J. Peter Grace, head of President Reagan's Private Sector Survey on Cost Control, launched a $3.1 million advertising blitz to promote hundreds of federal budget-cutting measures that his team of corporate executives recommended last year.

The bill for the 14-month television-and-print campaign, which debuted on New York, Chicago and Los Angeles stations shortly after the polls closed, will be picked up by W.R. Grace & Co. The company said it timed the announcement in the hope that the ad campaign would be seen as nonpartisan, while still taking advantage of election-night visibility.

Presidential advisory panels normally don't go to such extremes to publicize their findings, but the Grace Commission has been atypical from the start.

Appointed by Reagan in June 1982 to root out inefficiency in government programs, Grace and his 160-member executive board brought in hundreds of corporate accountants, lawyers and financial analysts to pore over the files of virtually every federal agency. The results were distilled into nearly 2,500 recommendations that Grace said could save the government $424 billion over three years.

The accuracy of the commission's figures have been challenged, but Grace's aides have defended their recommendations as worthy of consideration even if they result in a small fraction of the savings.

Grace is clearly eager to resume the cost-cutting effort after a campaign season full of promises to cut the federal deficit without touching anything important.

He recently has stepped up his efforts to sell the public on the panel's recommendations. He has published a book on his experiences with the cost-cutting panel and has joined forces with the conservative Citizens for America to lobby Congress, which would have to approve most of the measures.