A yagi (diretional) antenna is good, however once you are within 0.5-1.0 MV and greater signal strength (within a few miles of the pirate station) the signal is more than saturating the radio reciever and moving the antenna in different directions won't make a difference. You would be better with a signal strength meter connected to the antenna so you can see some type of increase in the field strength.
Ramsey Electronics makes a doppler direction finder DDF, but it is difficult to use and it is designed to work up in the 140MHz range which isn't the FM radio band. We've used one, but it needs to be modified according to the equations provided in the design notes to work for the FM band.
There are some commercial doppler units that the FCC and FAA use. These are expensive, $1000-$2000 plus. A field strength meter works good and you just need to criss cross through the signal center a time or two to find the pirate antenna.
Another alternative, we have used many times, is to start with 1) a listener complaint to get the general part of town, 2) drive toward that area repeatedly scanning back and forth on the car radio and note where the radio begins locking onto the pirate station consistently. Then drive through that area till you can't lock onto the signal anymore. Double back to the center and 3) use a pocket radio, without an antenna, inside the car the reception will be minimal until you are within blocks of the station 4) find the center of reception again and then 5) cover the pocket radio in foil to limit the reception even more and then narrow in on the center. 6) Look for an antenna on a mast, once you have found it, verify it by 7) wrapping the radio completely in foil well - you will only get reception from the street in front of the crime scene.
In 99% of the pirates we have found, they are renting space in a commercial office building. Write down the phone number off of the "for lease" sign in front, or look up the address on the county's property tax appraiser website to determine the owner. Notify the owner that there is an illegal station operating from the property and that you have already notified the Federal Communications Commission and that you will pursue a civil suit for damages against the property owner if the antenna is not removed within 12 hours. Most of the time the landlord is not even aware that a tenant has put an antenna on the roof of their building and they'll take care of it immediately.