I would start from power consumption. It's a little unclear to me if you're talking about electron use in all of the components or the CPU, but since 3 Volts was used, I'll take the discussion to be limited to CPU. Let's say 30 W of power consumption.
$$I=\frac{P}{V}=\frac{30 W}{3 V} = 10 A$$
Now, for the electron flux (denoted $\phi$), we can just divide by the electron charge.
$$\phi = \frac{I}{e} = \frac{10 A}{1.9 \times 10^{-19} C} = 5.263 \times 10^{19} \frac{1}{s} $$
This is the number of net electrons passing the power supplying terminals. I've heard that the drift velocity of electrons is 1000 to 10,000 times less than the actual velocity of electrons. So... you could just multiply the above number to get $5.3 \times 10^{22}$ electrons participating in current passing through a cross section of the terminal wire, but this is unsatisfactory for a number of reasons.
Firstly, at any given point in your circuit there are lots of electrons passing every which way. Why bother counting these unless we're talking about the role of conduction band electrons in temperature itself? Additionally, even if we restrict ourselves to the net electron flux in a wire - there are LOTS of wires in the CPU, and the passing of electrons through the terminal wire is not exclusive with electrons passing through other points, and a CPU is a very complicated machines with many many routes. In order to obtain more objective measures, you might want to look at the rate at which electrons bleed off, or loose voltage in some way (this would be the most analogous to a computer "using" those electrons). Otherwise, there is a virtually unlimited number of net electrons passing through all points in the circuits.
I also don't mean to present this as a very thorough answer, but at least I've taken a different approach to get the same order of magnitude of electrons that the question had. These numbers are just generally representative of a few Amps, but one needs to be specific in labeling what that number represents.
