What Else do I Need to Make a Speaker Work in These Conditions?
The "motherboard speaker" in PCs has usually been just a bare dynamic speaker. As such there is no need to create a circuit for it, just connect the two speaker wires directly to the appropriate connector on the motherboard.

Certainly one could create an amplifier circuit for it if one wanted, but the sparse use the speaker gets along with the requirement to power the amplifier itself makes such an endeavor essentially pointless.