I wouldn't recommend that, for a couple of reasons (but I have not had any kids).
Keeping track of your fertile and less-fertile days is very unreliable because you are only less fertile at certain times of the month. You can still get pregnant if you are not using any form of protection.
Secondly, think of what it will do to your sex life and how limited it will be. I think it would be a mood killer if you only had certain days where you would have sex.
If you really don't want to use birth control, use condoms, but have an emergency backup (like the morning after pill) with you at all times just in case the condom breaks.