I have never had any severe problems with curling of the film base in the 35mm format, but I have had some film in 120 format go nuts on me. Some of them, Foma for instance, curl like mad no matter what the weather is like (high or low humidity), others like Agfa and Ilford, dry almost perfectly flat. What I have discovered, however, is how effective it is to dry the film in a smaller room with a steam humidifier in it. Now that gives super-flat negatives.
I agree the weights help a lot, and that putting the film in Print File sleeves and putting them under weight is another good suggestion. I'm actually surprised that Kodak curls like you say. That's a brand that's been particularly good in that regard (to me).
Now I have a room that's very cold in the winter, around 50*F. So the film still dries slow enough that I don't have to worry about curling, unless it's a quality of the film base itself, like 120 Foma film.