The most common and most likely answer to this crossword clue is the 8 letter word MOURNING
I've seen this crossword clue in The Daily Telegraph
M
O
U
R
N
I
N
G
More clues leading to the result MOURNING
Buddy explains!

The straight part
We know that "grief" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case grief = mourning.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does MOURNING mean?
MOURNING verb- To express sadness or sorrow for; to grieve over (especially a death).
- To utter in a sorrowful manner.
- To wear mourning.
- The act of expressing or feeling sorrow or regret; lamentation.
- Feeling or expressing sorrow over someone's death.
- The traditional clothes worn by those who mourn (in Western societies, typically coloured black).
- Drapes or coverings associated with mourning.