Hi Johan,
It's usually termed seasoning rather than curing, in the UK at least. If you go back a bit there were no kilns, and old timber works fine, so it's certainly not essential. I don't have a kiln either. The main impact is time, as air drying takes a lot longer.
You need to stack the timber off the ground, in neat, vertical stacks, ideally in a shady place but where plenty of air can get round it.
In between each board you need to place a row of 'sticks' about 2' apart. These should be sawn to an even thickness, and should be seasoned before use or you'll get blue stain where the sticks touch the boards. Breaking up old pallets gives a good source of these for modest quantities. The sticks should be placed vertically above one another in successive boards to avoid putting a bend into the boards. The thickness of the sticks can be used to slightly control drying rate. If you're milling in autumn you can use thick sticks - about an inch. If it's spring you can use thinner sticks, say half inch, depending on how wide your boards are. Try to avoid milling over summer if you can as it risks rapid drying and the boards cracking. I know this is in a kiln, but the pictures linked from this thread show an excellent example of sticking:
http://arbtalk.co.uk/forum/milling-forum/48467-huge-tree-milled-fens.html
Once stacked, with something to keep the rain off the top (but don't cover the sides or it won't dry) you then wait. And wait..... You need about six months per inch of thickness for air drying. It will take longer in a wet climate.
A lot then depends on what you want to do with it. If it's rough construction, say shed framing, you may not bother to wait that long. If it's floorboards or other indoor work then you need to minimise movement. The easiest way is to bring the timber into the final environment before final finishing if you can.
As an example, when I did the floor in our old house I had no choice but to mill offsite in the middle of August in a heatwave. Boards were taken straight off the mill and piled in the shade with no sticking and I stuck the slabs from the sides over the top. The following day I picked them up in a covered lorry, and unloaded after the heat of the day had gone, around 8pm again into an unsticked pile in the shade and covered it with a tarpaulin. I then stacked them over the following couple of evenings, on the North side of a hedge, using half inch stickers and keeping a top cover on the stack (and the unstacked boards under tarps). It was so hot that I even side-shaded with some loose bits of ply, which I left covering during the day and took them off at night - I stopped this after a couple of weeks when the heatwave was over. The boards were milled at 1.5", so I let them season for around 9 months. I then rough sawed out the floorboard widths and brought them into the house, where they stayed behind the sofa for a few months. The following summer I planed them up to 1" thickness and jointed to width, then fitted them over the autumn. No movement at all.
I hope the above helps.
Alec