Being bit-pay.com a US based company I'm not so sure they will be the right ones to process payments to a pharma site.
https://bit-pay.com/acceptableUse.html

Also, the beauty of Bitcoin is that they don't need 3rd parties.
They can just generate a thousand addresses, save them in a database, and have their usual developer make it in a way that it shows a unique address to each order. Probably checking the payment completion is something they can do by themselves, without the need for automation.

Or they can just use walletbit.com and their handy API. https://walletbit.com/docs/api
Any PHP developer will easily integrate that, even payment received confirmation.

Ofcourse using their own Bitcoin client is preferred as they will not have to pay fees.

Big warm welcome to theswisspharmacy.com!  For anyone visiting this thread, I will vouch for them.  I have ordered from them in the past and know for certain that they have excellent customer service and high quality products.  They run a tight ship.  I came to them after having trouble with another internet pharmacy and I have been very happy with my choice.

Please help them in any way you can- especially allow them to earn your business by buying your medications from them with bitcoins!

I think what they need is a company that does an "original credit transaction" as described here:

http://www.forbes.com/sites/jonmatonis/2012/05/07/bitcoin-funded-debit-cards/

Also it might work to sign up at CryptoXchange and sell your bitcoins there, then have them wire the money to your bank.  I think they have pretty low fees for that type of transaction.

I spent some time to find out how new potential clients that I can convince to join the Bitcoin community charge their accounts with their credit card.
mtgox.com and aurumxchange.com seem to only allow them to charge with a wire transfer but not a VISA card. (In this case my clients could also transfer directly to my account….)

Where can my clients create an account and buy Bitcoins with a MasterCard/ AMEX?

+1 - this is what I do at casascius.com - use pre-generated addresses - this also keeps me safe from hackers because the bitcoins aren't ever on my website, nothing for them to steal, far less incentive for them to hack it.

Simply have the website e-mail you a link to http://blockchain.info/address/PUTTHEBITCOINADDRESSHERE so you can click it and directly see whether the amount has been paid.

Now that is a real a drawback!
I understand the problem though, just wonder how to convince new clients to go through the process of becoming a new Bitcoin member, if they can pay me directly with a wire transfer.
Looks like offering Bitcoins as a payment option on my website, is only convenient to existing Bitcoin users, or users that anyway want to use this option for other payments  as well.

1- Browsing internet i see this site: www.buybitcoin.com/

There are places to buy bitcoins with credit cards and other conventional methods, but they must charge more than the exchange rate in order to cover the risk involved. I didn't mention them since it sorta defeats the purpose of using BTC, IMO.

I would send people to http://www.weusecoins.com to learn about Bitcoin and where to get them.

Don't worry about asking people to learn about Bitcoin to use the pharmacy.  Just accept Bitcoin for those who already have Bitcoins and who know how to use them.  The number of people with Bitcoins is growing every single day.  If people ask, weusecoins.com is there to teach them, but people learning about Bitcoin for the first time are not likely to pay for an order with Bitcoin today, it may take them a day or a week or more to get bitcoins, so just take their credit card.

The appeal of buyers using Bitcoins for pharmacy is privacy and security.  As a buyer I don't have to worry a seller will share my credit card number, or charge me more money than authorized, or that my bank's loss prevention department will start inconveniencing me due to unusual activity.  My potential loss is simply limited to the amount I sent in bitcoins.

The appeal for sellers accepting Bitcoin is irreversibility of the payments and the inability for somebody else to shut your payments off because they think you shouldn't be selling whatever you are selling.