They believe many of the new measures, while costing business millions of pounds to implement, will not achieve their aim of preventing dishonest directors abusing their power over their companies.
If the rebellion continues to gather strength, an increasing number of companies will feel able to reject Cadbury in the corporate governance statements they are obliged to make in their annual reports. That could set back by years the attempt to reform the way companies are run in Britain.
Mr Melville-Ross, who is leaving his job as chief executive of Nationwide building society in August to head the IoD, said: 'The Cadbury report is too prescriptive, too limiting. We want to tell companies that they should not be frightened to say they are not following Cadbury and explain why.'
One example, he said, was that many companies simply did not need the special audit committee prescribed by the Cadbury code.
Sir Adrian Cadbury's report, published last year, lays down a code of best practice for companies in such areas as the appointment of non-executive directors, the remuneration of executives and financial controls. Although the report itself is not legally binding, companies are obliged by the Stock Exchange to state in their annual reports how far they have implemented its recommendations.
The IoD's attack is the most outspoken so far by any official body representing business interests. Although company directors have often grumbled about the plethora of new rules, few have complained publicly until now.
'The huge number of non- statutory rules is making it difficult for directors to do the job of looking after the interests of their shareholders and directors,' said Rudolf Agnew, newly appointed chairman of Lasmo, the oil company fighting a bid from Enterprise Oil.
Sir Christopher Hogg, chairman of Courtaulds, Reuters and SmithKline Beecham, believes that it would be a mistake if British companies ever had to comply with regulations that had become as complex and bureaucratic as those of the Securities and Exchange Commission in the United States. Companies might cease to take risks and 'play the process'.
A large institutional investor said: 'Companies can spend all their time ticking the right boxes to show they've complied with the nitty-gritty of Cadbury, but it's rubbish really. It distracts from how well a company is actually doing. There is a danger of losing the big picture.'
Directors have so far held back from open criticism of the corporate governance rules, partly out of fear of sounding politically incorrect. They are also afraid that if they do not comply, their shareholders may wrongly conclude that they have something to hide.
However, many now fear that there is a danger of repeating the fiasco of City regulation in the 1980s. The broad principles of regulation embodied in the 1986 Financial Services Act commanded widespread support. But they were turned by the Securities and Investments Board, led by Sir Kenneth Berrill, into thousands of detailed rules so onerous they had to be shortened and re-written three years later.
Sir Adrian Cadbury was not available for comment.