I also do not have a credit card, but I do have a Visa debit card, so it is possible to do this without a credit card.
They also have a postal order and Cheque payment scheme, but I'd guess you would be too late for that.
However:
The issue of permitting only a single supplier of something as important as the ticket payment I agree should be completly illegal.
I also fail to see the point, it can not work in anyone's favour, not the Olympic organisers, as they will have lost hundreds of thousands of ticket sales, the punters who feel left out, or even Visa themselves as this will have cost a fortune for very few new customers and scored zero on the popularity meter.
What will be interesting is to see if they have actually sold all the tickets. Whilst popular events will of sourse be oversuscribed manyfold, I'll bet the 30,000 tickets to see say the heats of the Fencing on a wet Monday morning at the Dome, or Azerbijan Vs. Montenego Football at some obscure northern stadium on a Thursday afternoon will not be sold at any price.
No doubt Seb Coe will come out with a statment (which was written weeks ago) to say that the tickets were 'massivly over subscribed' but that was just the mens 100m, not every ticket sold.
As an aside, I will make a point of wearing the T-Shirt of the company that does not have the concession on the food/beer/money/photography or whatever. (Remember the girls in the orange skirts who got thrown out of the ground at the Dutch World Cup match? I therefore remember Oranjiboom beer, not whomever was the beer sponsor - no idea who it was)
Perhaps we should start a movement... Ask everyone going to the Olympics to wear the shirt of a non-sponsor!