I find that they often do a fine job, especially in the early game. Before you've built a lot of improvements, choosing which tiles to work is basically a no-brainer.
I get annoyed with tedious micro-management, so I'll use the governor in some cases to save time. Usually I turn on the governor, then tweak his choices a bit myself. For example, turn on governor, emphasize food. Turn off governor. Add specialists. Why manually assign workers to the highest food tiles when it can be done very quickly by the governor? (When you add specialists they will automatically come from the lowest-food tiles.)
The "turn off growth" option (or whatever it is) is useful a lot too - again, mostly in the early game. Often times I am at my happiness cap and need to maximize production, yet slavery is not appropriate. In this case I will usually use the turn off growth button. (Be careful using this though. Don't forget and leave it on, especially if you build a Worker or Settler.)
Also remember that even if the governor is off, the "emphasize X" option will still apply to any new population. You can use this as "insurance" in case you miss that extra pop and forget to manually assign him to a tile.
It is certainly preferable - even required - to manually manage your cities if you want to move up in level. But that doesn't mean you can never use the governor: in new cities, or to save time (especially in multiplayer.) What it comes down to is whether I can accomplish the same thing by using one click (the governor) that I could by manually going into the city and clicking many times. If I know the result will be the same, I'll use the governor.
