Electric circuit AC plug to DC Design Help
No, that circuit won't work. You have a closed circuit for the lamp (the right loop), but the motor and resistor are just shorted; the loop they make doesn't have a voltage source in it.
The solution is a DC power supply, which unfortunately consists of quite some components.

First you'll have to lower the 120V/230V mains voltage to a more workable voltage to go to the 3V. That's what a transformer does. Next, you'll have to rectify the low voltage AC that the transformer outputs.  



The bridge rectifier takes care of that. The smoothing capacitor is needed to remove the bumps from the rectified voltage. You then have a more or less smooth voltage, but it will vary with varying mains voltage and with a changing load of the motor. If you need a clean 3V DC you'll have to end with a voltage regulator.  

If you don't need the regulator you'll need a 3.6V transformer, which will be hard to find.
A 5V transformer will give you 5V DC across the capacitor. Replace the 7805 in the schematic by a 3V LDO voltage regulator to drive your motor.

edit
jippie makes an excellent suggestion.  



Wall-warts like this one exist that output a regulated 5V DC. This includes everything in the schematic. If your motor accepts 5V you can directly connect it to it, if it has to be 3V you'll need the LDO voltage regulator.