The answer to this question is "it depends on your definition of dangerous."
If you are asking whether it is dangerous in the sense of whether it is likely you will be kidnapped by a terrorist and have a toe chopped off and sent to your spouse in an envelope, then no, it's not dangerous. Ever since the bombings in 2003, the country has cracked down on terrorist activity and I am happy to report that I have never felt unsafe here. I am told things did get bad for Westerners just before the bombings and there was some harassment going on, but these days I would be very surprised to hear of something like that happening.
If your version of 'dangerous' entails lashings by angry bearded arab men yelling infidel at you because a button is broken on your abaya, then the answer is also no, it's not dangerous for Westerners. The Muttawa once roamed free in this land, travelling in packs, some say carrying sticks, and when there was a full moon out they would grow fur and claws and howl while they feasted on the flesh of... oh wait wait...I'm getting mixed up here. It's hard to separate myth from reality, past from present, rural vs. urban with the things you hear about the religious police. For the most part the main kind of trouble they will currently give a Western woman is for not having her head covered. I believe locals and muslims are under more scrutiny, but that being the case I would still not recommend Westerners to be alone in public with the opposite sex if they are not married or related to them.
If your definition of 'dangerous' is petty crime, then yes, Riyadh is dangerous in the same way Toronto, LA, New York, and other countless big cities are dangerous. Add the high proportion of youth and their matching unemployment rate, you can sort of get an idea of what type of trouble brews here. People occasionally get carjacked, and muggings occur, though they usually target working class Filipino and Indians who have no recourse. If anyone tells you that the benefit of coming to a holy land is that the cities are very safe, you have my permission to laugh a little. I say this because hubster told me that before we moved here and we now have a good laugh from time to time about it.
Lastly, if your definition of "dangerous" is the most insane and violent road rage you've ever seen in your life, then on very rare occasions Riyadh also fits the bill. I'm only gonna say this once: if the other driver looks crazy, he probably *is*. In this specific circumstance, resist the temptation to get out of your car to engage the angry man who has just cut across three lanes to slam into you, because although he may look like a skinny nothing, he probably has a lead pipe under his seat saved for special occasions like this.
Though the majority of the country is perfectly fine with Westerners being here, there's still a segment of people who resent us and what we stand for, and it's best to remember that when you're out and about. They all speak English and can understand disparaging remarks, so save your frustrations for your house or car. I've never felt unwelcome or truly unsafe here - but if you go looking for trouble in Riyadh, you can find it. As a woman and Westerner, I am out and about regularly on my own and do not think twice about it. Riyadh is as safe as any other metropolis, so don't worry too much about it!