My bank is not on Coinbase's list and I was able to add it. This takes a couple days.
Choose "Settings" on the left navigation panel, then "Payment Methods" from the choices across the top of your screen. Next, click on the "Add Payment Method" button on the right of the screen. A dialog box will appear.
Within that dialog box, scroll all the way down and choose the box which is labeled "Other Bank." The dialog box information will change.
From here you begin to enter your banking information. Within a day or so, Coinbase will deposit a small amount into your account, and then withdraw it. When this occurs, you will need "verify" the deposit by entering either each deposit amount, or the total of the deposit (I forget which) in a verification screen. At this point, you can initiate a deposit from your bank.
Be advised, however, each time I have done a deposit, it is one week before i can use the funds. Coinbase tells you 5 business days, and when you see the message, it will show 6 days. BUT, if I initiate a deposit on a Tuesday, I cannot use the money until the next Tuesday.
The quickest way is to be verified and use a credit card to deposit funds. I finally became verified, but I have yet to attach a credit card. For some reason, Coinbase cannot verify my address to my credit card. You may not have the same problem, lots of people don't. But, you can only try once every 24 hours to do this.
I hope this helps you if you decide to continue with Coinbase.
I've chosen to use TradeQuick, however, as @williambanks suggested, as I've had so many issues with Coinbase.