Note: While this isn't a complete answer, it turns out that your intuition is mostly right.
Wind currents would, generally, go from the day side to the night side. This depends, however, on a lot of factors. Such factors can be temperature of the star, landmasses, density and composition of the atmosphere, oceans, and others. Cornell University did a neat study on tidally locked planets, if you would like to take a look. They even address Coriolis Effects! As a simplification and summary, the faster your planet goes around your star, the more your winds get mixed up.
We know that, eventually, the air from the cold side will "want" to go to the warm side, due to pressure, gravity, and diffusion. This may form the equivalent of Hadley, Ferrel, and Polar Air cells, except it involves air going from the day side to the night side instead of equator to pole. It turns out modern simulations find that such circulation would actually keep the night side quite warmer than previously thought. See the wikipedia article about the Habitability of Red Dwarf Systems.
The Air Cells
It would be generally assumed that the air cells on our tidally-locked planet would look like Hadley/Ferrel/Polar Air cells going from the day side to the night side. Some people think there would be areas of permanent rain where hot/cold air meet, and that there would at least be a ring of habitability somewhere around there.
Climate Models Are Complicated
You should therefore recognize that this answer has some assumptions built in. The Atmosphere needs to be neither too thick or too thin. It appears that oceans ought to exist to make it habitable. Many studies don't actually look at wind, but rather heat transfer or other effects. Your question is closely tied to exoplanet study, and it is a huge field.