Is it possible to have a SUMIF total the values in multiple sets of ranges?

EG I currently have the following formula-

=SUMIF('Sheet2'!$A$5:$C$289,'Sheet1'!$A$4:$A$42,'S heet2'!$B$5:$B$289)

This formula checks if there are values in Sheet2, Column A that match the value of a specific line in Sheet1, Column A.

Sheet 2 is arranged like this:

Code:
A         |  B
Client 1 |  $4
Client 1 |  $3
Client 2 |  $65
Client 1 |  22

If there is a match, every item in column B that matches that criteria is added-up into one nice, neat cell.

However my data set has changed so that I have to sum-up several non-contiguous cells from the following data set:

Code:
A         |  B  [.....] | F
Client 1 |  $4         | $23
Client 1 |  $3         | $3
Client 2 |  $65       | $23
Client 1 |  22         | $83
For reasons beyond my control (I blame government), there's no way I can change these datasets to have the ranges of numbers to be summed-up so that they are grouped next to each other.

I can do multiple SUMIFs in one cell, but I have to add-up six different non-contiguous columns so the formula is really huge.

Any Excel wizards have any more elegant solutions? I should also mention I'm doing this in Google Docs, but the formulas are mostly the same.