The changes apply to England and Wales and are planned to come into force from 1 April. No additional funding will be forthcoming.
'The current system of allowances is too prescriptive and places an unnecessary burden on local authorities,' Mr Curry told the Commons.
'The changes we propose will remove that burden and will encourage people with a wide range of skills to serve as councillors.'
Ministers have accepted arguments by the associations that the level of basic allowance should not be laid down by government.
Association of Metropolitan Authorities secretary Rodney Brooke welcomed the proposals as an important piece of deregulation for local government.
'For years there has been discontent that members have been giving up a great deal of free time and getting minimal reimbursement of their costs while others, such as those on quangos, have been paid more money than members could dream of for much less work,' he said.
He regretted there was no movement on the impact of allowances on benefit payments, and hoped the DoE would still be willing to explore reimbursing employers for time members spent away from their jobs on council business.
'Members do not wish to jeopardise their jobs or pensions,' he said.