The trio - from Canada, Norway and the Philippines - appealed for help from the Philippine and Canadian governments in a video uploaded to YouTube Tuesday and reported by SITE intelligence group, which monitors jihadist media.
"To the Philippine government: Please stop shooting at us and trying to kill us. These guys are going to do a good job of that," Robert Hall, a Canadian, said in the video.
He said the kidnappers had told him to tell his own government "to meet the demand" -- though he did not specify what the demand was.
The southern Philippines-based Abu Sayyaf announced last week they had killed a Canadian hostage, retiree John Ridsdel, as an April 25th ransom deadline lapsed.
Norwegian Kjartan Sekkingstad and Marites Flor, Hall's Filipina girlfriend, also pleaded with the two governments.
The video showed them sitting on the ground, beneath what appeared to be a coconut plantation, with six gunmen standing behind them.
The Norwegian said that "if the demand is not met we will be executed like our friend John was a few days ago".
One of the gunmen then warned Manila to heed the "lesson" of Ridsdel's killing and avoid delays in negotiations.
Philippine President Benigno Aquino has vowed to "neutralise" the gunmen and his government has said it will not negotiate with a group it considers a terrorist organisation.
Canadian Prime Minister Justin Trudeau also said -- after announcing Ridsdel's murder -- that his government would not negotiate with the kidnappers.
Aquino said the three hostages were being held on the Abu Sayyaf stronghold of Jolo island, more than 1,000 kilometres (620 miles) south of Manila.
Ridsdel and the three were snatched from a resort elsewhere in the southern Philippines in September last year, with the kidnappers demanding millions of dollars in ransom.
On Sunday the gunmen freed 10 Indonesian sailors they had abducted at sea five weeks earlier.
But they are also still holding four Malaysian and four other Indonesian sailors, as well as a Dutch birdwatcher abducted in 2012.