I'm not sure about sieve, but grieve comes from the Old French 'grever': to burden and oppress. The meaning of mourning (usually because of death) was added to by 'The Reavers' - people who lived on the borders between England and Scotland many centuries ago and carried out raids into both countries, stealing cattle and killing people. Hence to "bereave" = to deprive someone, especially by death. Ignore all the spelling differences between these words.
The nearest usage of grieve to the original 'burden' and 'oppress' is in phrases such as "I'm getting so much grief from my boss!" "This car has given me nothing but grief".
EDIT
I was curious, so I looked up sieve.
It's Old English 'sife' which comes from Proto-Germanic 'sib' or 'sibi' and is akin to the Dutch 'seef'. I, personally, don't find that very illuminating, but I've noticed that many words have migrated from 'f' to 'v' because of the pronunciation effect of different vowels either before or after them.