What you need is a Brown Dwarf. Technically, these are not quite low mass stars, but a bit bigger than (say) Jupiter. They give off heat, although not much. Your planets would be a great deal darker than Earth, and they would have to orbit closer (radiation issues), and your system would last for less time than a main sequence star, but it's possible.
Ideally, the brown dwarf would have formed around a main sequence star at about the orbit of Saturn, allowing the main sequence star to contribute more light and heat to your system. This would mean that your planets (moons in this model, really) would have more light and heat introduced to their systems on a periodic basis, so night and day would be very confusing out there. I'd also add that unlike Earth, you'd probably WANT a great deal of greenhouse gas to trap what little heat reaches the planet into its biosphere.
Think Venus around a much heavier Saturn. (Saturn is only about 30% the mass of Jupiter, but nearly as big.)
As for the planet/moons not being tidally locked, again possible, but depends on the forces in play during creation. Tidal locking is common because it's the least energetic configuration. Put something in play that strikes the forming planet with a glancing blow or two, and you'd probably get rotation.
Ideally, you need a moon for your moon to generate a magnetic field via a spinning core; your brown dwarf is going to release most of its heat as radiation I suspect. Great for heating the surface, but pretty bad for survival rates of humans.
I must say; this is not a nice planet to live on. You'd probably need a human adapted to high CO2 levels (either that or you dump a heap of methane into the air), a form of fungus in lieu of conventional plants (low light means less photosynthesis) and good night vision.
Another option which goes against your OP would be a low mass star, but for now I think a Brown Dwarf meets the spirit of your question (just) and solves for the needs you have.
Of course, if you really want a gas giant, just move Jupiter closer to the sun (say Mars orbit) and the latent heat coming off it (in conjunction with the heat from the sun) may also provide you with a livable planet. Even though Jupiter and Saturn are 'planets' they still give off some heat. Generally though, it's not enough to turn the tide in terms of sustaining a biosphere by itself.