Definitely should be "my grumble" . 'me' is first person singular used as an objective pronoun, i.e. a pronoun which is the object of a sentence as in "He gave it to
me".
"My" on the other hand is a possessive (belonging to) adjective, as in "This is my book". On the other hand you could say "This book belongs to me", in which case it is the first definition. After all this is a "Grammar" thread!
On the other hand colloquially "me" has been used as a replacement for 'my'
Look no further than Australian Frequent Flyer threads for solecisms!
Picked this up in another place; eat your heart out grammarians! (my BOLD).
"The object of the preposition
with is a non-finite clause, and as you know, there are two potential versions of that clause.
One with the
genitive pronoun as subject: "
my presenting a final thesis".
And one with the
accusative pronoun as subject: "
me presenting a final thesis".
In today's English, both versions are grammatical, though the
my version is probably a little more formal. But I would advise you to go with the one that sounds most natural to you. Since the pronoun immediately follows a preposition,
me will probably be preferred by most speakers."