The Murrays have never lost a Davis Cup doubles rubber but crashed out in the first round in London 2012, with Andy going on to win singles gold.
They also paired up in Beijing in 2008, suffering a second round loss.
"It's four years to sit on that result, for me personally anyway, so we will be trying our best to do as well as we can and make up for London," said Jamie.
"We want to do really well in the doubles and we're both in the peaks of our careers.
"London was a big disappointment for us in the doubles front, although Andy did amazing to win gold in the singles."
Jamie, 30, is ranked the world's number one doubles player, with his younger brother the number two in singles.
Having reached two finals in 2015, Jamie gained his first Grand Slam men's doubles title in January, winning the Australian Open alongside partner Bruno Soares.
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And the Scots-Brazilian pair reached their first Masters final this month, losing to Pierre-Hugues Herbert and Nicolas Mahut in Monte Carlo.
"A lot of good things happened for me in the last six to eight months," said Jamie on a visit to his hometown of Dunblane.
"I've got some big tournaments coming up in Madrid and Rome in preparation for the French Open then straight into the grass courts at Wimbledon.
"So there are a lot of good things coming up and I'll try my best in each tournament."
Last year, Andy, 28, led Britain to their first Davis Cup triumph for 79 years, winning all of his singles and doubles contests.
The title defence continues with a trip to Serbia in July but, with the tie coming just five days after the Wimbledon final and on a clay surface, Andy's participation may be in doubt.
The 2013 Wimbledon champion says that he plans to play but it may depend on how he fares during the short grass season.
"He is our best player by a country mile," said Jamie. "He carried the team to the title last year.
"The summer this year is really tough with the Olympics thrown in there and, for us to play Wimbledon and to play a few days later on clay in Serbia, is not easy.
"Then it's straight to America and the Olympics.
"But we'll see what happens. For him, as long as he's feeling good with his body and his fitness then there's a good chance he'll play."
Jamie Murray is targeting Rio doubles success with brother Andy to make up for past Olympic disappointments.