Can I meditate while lying on my back? Is lying flat on my back an acceptable meditation posture?  Why or why not?
You can, but it's not recommended.  Many people come to meditation with the notion that it is a relaxation exercise.  With that preconception, it's no surprise that so many wonder why we just don't meditate lying down or in an arm chair.  There may be some kind of benefit to this sort of mediation.  Whatever the benefits, however, they do detract from the kind of meditation taught by the Buddha.  

There is a very strong connection between the body and the mind.  The condition of one really determines the state of the other.  To meditate properly (from a Buddhist standpoint), it is very important for the body to be in an upright and invigorated position.  When we lay down or sit in a comfortable chair, it is all too easy for our minds to go slack and lazy.  These sorts of positions effect our concentration and makes it all too easy to fall into sloth and torpor.  You don't want that.  To hold the body upright is to hold the mind upright.  What you want is a state where you are as awake (from a physiological standpoint) as you are going to get.  Discomfort, even a little pain (note - little!), is very conducive to raising your attention and applying it on the meditation object.