I am going to give some details because the devil is in the details and you need to know how difficult it is to handle what you are proposing.
As an example:
Assuming you pass through immigration separately - separate lines for GCC nationals. You can travel around like that but in absence of her having an American passport; the whole ruse will fail if you run into any brush with the officialdom e.g. a car accident or whatever. In that situation, both of you will be in serious trouble. Besides, you would stick out like a sore thumb anyway i.e. an obvious Arab looking girl with a non-arab guy (my assumption - correct me if I am wrong) - will attract unnecessary attention. Then unless you stay with her parents, you won't be able to stay together in hotels without providing identity documents along with proof of marriage. You will not be able to provide that. Hotels photocopy passport and visa on check in - the moment she takes out Saudi passport or ID, the problem will start.
So the only way this "really" works in a risk free manner is for her to get American nationality, renounce her Saudi nationality (if it is not stripped by the government first - as they don't allow dual nationality), get a visa and then come with you. Don't know if she would be willing to do that.
Beyond dowry, you need to find out if they will ACCEPT their daughter marrying a non-Muslim foreigner or even for that matter, a foreigner. The majority of the general populace wouldn't even consider it - dowry or no dowry.
So in short, if you want to maintain the association with Saudi and the family here; you would need to go through proper channels by applying for permission and hoping for the best. For that, you would need her family on your side 100%.
On another note, if you proceed with this, develop a thick skin. I have friends who are in similar situations e.g. Pakistani guy married to a Saudi girl of Pakistani origin, through proper channels, wife speaking fluent Arabic etc and they get hassled at immigration almost every other time with some officers routinely making it difficult by asking the girl why she married a Pakistani. The kids and husband don't have Saudi passports obviously, so most times, just to make a point, they let the woman through the GCC national line while they send the rest of the family to the busiest most crowded line for expats even if she is ok to wait with the family in the expat line. Other issues in the country as well when moving around together. Unfortunately, it is usually only an issue when the girl is Saudi - other way round is not a problem at all. Official permission will not change attitudes.