Griffiths, an MP in the traditional powerhouse of UK brewing, Burton, has been chair of the group for the previous five years - and has overseen and pushed for cuts to the beer duty escalator as well as campaigning for cuts to business rates.
A new chair will be announced following a meeting of the group next month.
The ALMR's Kate Nicholls paid tribute to the outgoing chairman’s "hard work and dedication to licensed hospitality".
Chief executive Kate Nicholls said: “I would like to thank Andrew for the dedication he has shown to the role over the past five years and his tireless promotion of UK beer and the licensed hospitality sector. Andrew has been a fantastic champion of pubs and bars in the UK and a vigorous champion of the great work we do.
“Andrew worked hard to bring down costs for pubs and bars, helping to promote investment and growth in our venues across the UK.
“The ALMR is looking forward to working closely with the new chairman to promote the UK’s licensed hospitality sector to achieve a fairer deal for hardworking businesses.”
BBPA chief executive Brigid Simmonds paid tribute to the "extraordinary achievements" of Griffiths.
She said: “What Andrew has achieved for Britain’s brewing and pub industry during his time as Chairman, as someone who when first appointed was quite new to Parliament, is quite astonishing and we all owe him a huge debt of thanks.
“He led the campaign in Parliament to remove the beer duty escalator, seeing three cuts and a freeze after years of sharp tax rises. He has worked for a better deal on business rates for pubs, and fought equally hard on issues that don’t appear so much in the news, such as costly proposals for tax stamps on beer, saving the brewing industry millions of pounds in the process.
“We wish him well, and Burton, as the country’s premier brewing town, is very lucky to have such a great champion of beer and pubs as their local MP.”
Mike Benner, managing director of the Society of Independent Brewers (SIBA) said: "Andrew has been a great champion of British beer and brewing and his achievements have been outstanding. The campaign for a cut in beer duty, which led to three consecutive cuts will go down in history as one of the biggest achievements for the sector and Andrew showed great leadership throughout.
"I'm sure he will continue to play a major role in supporting beer and pubs in the future and we thank him for his enormous contribution."