I have been to South Africa to dive with Great Whites and unfortunetly we missed out. We saw sharks from the boat and when the cage was put in to the water, the sharks were scared off. As mentioned before, in South Africa, it is only a day trip to see the sharks, so it is quite cheap. Don't let the crime put you off. It is a beautiful country. We did a 3 week overland safari as well, so all was not lost.
I have also been to South Australia and dived with Whites there. It was one of the best trips I have ever done. Big sharks in clear water. It was rather expensive ($2000), though this was a 4 day trip on a liveaboard. The food and service was fantastic. I am led to believe that cheaper day trips are now available, though you have more chance of seeing sharks with an extended trip.
With the exchange rate the way it is at the moment (US $1.00=AU $0.65), it works out rather cheap for Americans to travel to Australia. This might be an option for you.
Good luck in your quest to dive with Great Whites. Whatever the cost, it is well worth the expense as it is an experience you will never forget.