Kuehnen has been locked in a bitter row for months with world number 19 and Germany number one Philipp Kohlschreiber, who has ruled out a return for the team's first tie in 2013 against Argentina.
"In the past weeks I got the impression that I was missing the necessary support and backing from the German tennis federation (DTB)," Kuehnen said in a statement.
"So I have no basis of trust to continue working together and I end at this point the talks for continuing my work at the DTB," added the 46-year-old.
Kohlschreiber accused Kuehnen of failing to support him after former world number two Tommy Haas had questioned his commitment in February when he missed the first round 4-1 defeat against Argentina with a stomach bug.
The row had drawn in other former and current players with Kuehnen, who won the Davis Cup three times as a player, unable to resolve it.