The 26-year-old made three appearances for the Cherries in exactly five months at the Goldsands Stadium.
Joining on a free from Bundesliga's Augsburg at the end of August, Davids overcame injury to force his way into Paul Groves' plans, but failed to make an appearance under new manager Eddie Howe.
The Dutchman has signed a two-and-a-half year deal at Danish side Randers FC.
"I am delighted with the challenge in Denmark, where there is a lot of emphasis on speed and technique, just as in the Netherlands," Davids told randersfc.dk.