I've been using online billpay for years, at three different banks. Two were local (a bank and a credit union), and the other is ING Direct. I haven't had any problems with any of them that weren't self-inflicted (forgetting to enter the bill). The credit union's system is pretty clunky, but the other two are fine.
One thing to make sure of is to leave enough time for the bill to arrive, just like you would do if mailing a check. Just have the bill sent a week before its due, and you should be fine. I usually do this soon after I get the bill, so I don't forget about it. ING will actually receive bills from some companies automatically, if you wish. So all you need to do is go online and click pay, and it will know when the due date is and the amount to pay. For bills that have the same amount each month (mortgage, insurance premiums, etc.), you can set it up to pay automatically each month so you don't have to do anything.
Its a bit of a hassle moving banks, and reentering the account numbers, addresses, etc. Stopping a bill is as easy as clinking delete in the online system. My current setup is to have all my bills paid through ING, and my paycheck direct deposited. I can transfer money to/from my local bank in a couple of days if I have checks to deposit, or to use the local ATM.
I short, I would never go back to writing paper checks.