I have a question. How do you know that a place has good production levels? Because when I started ignoring the blue circles, my cities production went along much slower than before.
When I'm choosing city sites, the first thing I look for is a good food resource. Production tiles are usually low on food, which means you need a surplus from another tile so you can spare the citizen to work the hammer-heavy one.
So I try to include at least one food resource and at least one production resource in the fat cross. If I can get more of the latter, great, but I know I'll need another food resource to balance it out; I may therefore leave a second nearby production tile for another city.
Then I look at the other tiles, striving for the same balance. If there are some good production tiles (hills and forests, generally, especially combined with the plains overlay) I try to ensure there will be some that can be farmed for a surplus (grassland or even floodplains). Yes, watermills and workshops are an option--but not until mid-game at the earliest, so they don't get factored in to my site planning in the early game.
Now, here's the catch. Assuming I'm running a cottage economy (which I do most of the time), I want cottages on any grassland or flood plain tile next to a river to max out the +1 commerce from the tile. So if there are a lot of those in this production city's fat cross, well, I'm tempted to turn it into a commerce city instead. Production cities get farms on every tile that can have one, you see, to max out the production tiles that can be worked. My favourite water source for a production city, then, is a fresh water lake rather than a river, since I'm not losing a commerce bonus by farming tiles next to it. The map generator doesn't always cooperate here, of course.
Ultimately, I may have to adjust the city location, shifting it a tile in one direction or another to get the best combination of resources and regular tiles, and to support its chosen specialization. But to put it simply, a production city is all about balancing out the hammers and bread.