I too have a very mossy lawn which has been neglected for some years (we moved here last autumn) - I do not want to use chemicals to kill the moss as we have resident frogs, hedgehogs etc.
I am collecting the grass when we cut the lawn rather than leaving it on the surface, and making sure that we don't cut the grass too short..
I am raking the lawn vigorously whenever I get the chance (really good for toning the waistline and bingo wings and much more fun than the gym), gathering up the moss and thatch of dead grass and putting it on the compost heap, and I am spiking the lawn as deeply as I can with a fork to improve aeration and drainage.
In the autumn I plan to rake and spike thoroughly and then sprinkle the lawn with a mixture of sand and a variety of grass seed which will perform better in shade.
Fingers crossed that ought to improve things a bit 