Pet avoiding river in Greek valley (5)
I believe the answer is:
tempe
'greek valley' is the definition.
'pet avoiding river' is the wordplay.
'pet' becomes 'temper' ('pet' can mean a tantrum).
'avoiding' indicates named letters should be taken away.
'river' becomes 'r'.
'temper' with 'r' taken away is 'TEMPE'.
'in' is the link.
(Other definitions for tempe that I've seen before include "beautiful valley" , "peaceful valley" , "beauty spot" , "valley of great beauty" , "Grecian vale" .)