Why are you going there this month? It is Ramadan, no? Horrible time to be in Saudi Arabia. Business is very slow because offices are on a reduced schedule. I assume you're going there on business, as there is little other reason for an American to go there. It is an interesting country. I only went there to visit my father, an American working in the Kingdom. Not a great tourist destination...
As for getting there, BA
and Lufthansa offer the best connections to both Riyadh and Jeddah. AF
and Swiss offer service a few days per week. You can always take Saudi Arabian either from JFK
or any major European airport. Don't expect a drink.
If you have a contact in the Kingdom, he might be able to get you a cheaper fare by buying there. Such was always the case with my mother and myself on trips there.