Greek river bird (6)
I believe the answer is:
grouse
'bird' is the definition.
(I know that grouse is a type of game bird)
'greek river' is the wordplay.
'greek' becomes 'gr' (abbreviation for Greek).
'river' becomes 'ouse' (Yorkshire river).
'gr'+'ouse'='GROUSE'
(Other definitions for grouse that I've seen before include "No shooting it before 12 August" , "excellent in New Zealand" , "Bird - complaint" , "Bellyache" , "The complaint of a gamebird" .)