In place of the SUMIF(), you'll need to construct a SUMPRODUCT() to replace it.
Code:
A B C D E G H
1 Date Description Amount Code Start 1/1/2010
2 1/15/2010 Groceries $50.00 GROC End 1/31/2010
3 1/21/2010 Gas $22.50 GAS
4 1/25/2010 Groceries $50.00 GROC GROC $100.00
5 1/30/2010 Gas $22.50 GAS GAS $45.00
6 2/13/2010 Groceries $50.00 GROC
7 2/13/2010 Gas $22.50 GAS
The formula for H4 would be:
=SUMPRODUCT(($D$2:$D$100=G4) * ($A$2:$A$100>=$H$1) * ($A$2:$A$100<=$H$2) * $C$2:$C$100)
...copy that formula downward, it will adjust itself for each category.