It was the summer of 1987 and the Mariners had selected Griffey as the first overall pick in the First-Year Player Draft.
"I had just signed and was working out at the Kingdome when Dave came up to me, introduced himself and said, 'Hey kid, Dave Niehaus, you are going to be a good one.' That's all he said," Junior recalled.
Less than two years later, Griffey had become a fixture in the Mariners' lineup and Niehaus was yelling, "That ball will fly, fly away," at the end of many of Junior's at-bats.
They became best of friends and future Hall of Famers.
Niehaus broadcast all but about 100 all of the club's 5,385 games since being hired by the Mariners for their expansion season in 1977, and he was an integral part of the team's identification from the first day. Niehaus, 75, died of a heart attack on Wednesday night at his Bellevue, Wash., home.
"He is the Seattle Mariners," Griffey said of the Seattle icon.
Niehaus became as much of a Seattle fixture as the Space Needle.
Dressed in a tuxedo, he threw out the opening pitch at Safeco Field when the new park debuted in 1999, and was elected into the Washington State Hall of Fame in 2009.
He joined former first baseman Alvin Davis as the first two members inducted into the Mariners Hall of Fame in 2000, since joined by Edgar Martinez and Jay Buhner -- soon to be followed by Junior.
An obviously emotional Griffey said during a telephone interview that one of the things that stood out with Niehaus "was the way he treated everybody. There was not one person, from one to 25 on the team, that he didn't like. I don't think you can say a negative thing about that man."
Griffey spent two different stints with the Mariners, being drafted No. 1 overall in 1987 out of Moeller High in Cincinnati. After 10 memorable seasons with the Mariners, basically putting Seattle on the baseball map, he played nine seasons with the Reds and White Sox before returning to the Mariners prior to the 2009 season.
Niehaus was like a third father to him, ranking behind only Ken Griffey Sr. and Mariners president Chuck Armstrong.
Junior noticed in the spring of '09 that Niehaus was getting a little rotund around the middle and promised to get the broadcasting legend in better shape.
The news of Niehaus' passing hit Griffey hard.
"I have been a part of two guys now who mean so much to their organizations," Griffey said. "First, it was Joe Nuxhall [of the Reds] and now Dave that have passed away. Those two guys, Joe from the Reds and Dave from the Mariners, you can talk about every player who played in those organizations, but without their voices being able to relay to fans, the players were nothing.
"They always had a positive attitude and were able to laugh and joke, cheer you up when times were bad. I can remember early in my career, Dave would say something that would make me laugh. He would say, 'Good, I got you back to smiling.'
"This is tough. Like I said, he is the Seattle Mariners. He had the same impact in Seattle as Harry Kalas had with the Phillies, Harry [Caray] had with Chicago. There isn't a baseball fan in this country who doesn't know that Dave Niehaus is the voice of the Mariners."
And even though Niehaus is now gone, that will never change.
"No one in the organization, or any of the fans, wanted this day to come," Griffey said. "You never want to see this happen. It's a sad day for the people of Seattle and it's a huge baseball loss.
"You can talk about losing a player, but you are talking about a man whose voice speaks volumes. He will be sorely missed."
Asked about his favorite Niehaus memory, Griffey talked about the white shoes that the broadcaster wore.
"We kept telling him that he wasn't going to the prom again," Griffey laughed.