Do you earn rewards points by paying your monthly bills? If so, how?

I like to redeem credit card rewards for airfare and have done it many times but I would like to earn points faster by paying my utility bills, house payment, any thing I can with credit and then, of course, paying the card in full out of my checking account. Right not I use my bank's online bill pay and it is very convenient but I would give up some convenience to earn points. If you are doing this, are there any pitfalls or unforseen costs? What cards do you use for this?