How does ATC handle arriving/departing runways that cross?
Chapter 9 - Departure Procedures and Separation of the FAA Order titled "Air Traffic Control" covers the procedures that are used for aircraft departures, including those cases where they have intersecting runways.

There are numerous procedures that they follow that are specific to intersecting runways, including:


3-9-4 - LINE UP AND WAIT (LUAW)

If the tower has an airplane Line Up And Wait (taxi onto the active runway but not actually takeoff) because of another airplane that is landing or taking off on an intersecting runway, they must inform them of the other airplane and why they are not taking off.

3-9-7 - WAKE TURBULENCE SEPARATION FOR INTERSECTION DEPARTURES
3-9-8 - INTERSECTING RUNWAY SEPARATION

This probably covers most of what you are asking about, and has some great pictures showing when they are allowed to clear another airplane for takeoff in different situations.  For instance, they can't clear an airplane for takeoff with another airplane landing on an intersecting runway until after they have passed their runway.



Note that there are similar procedures in Chapter 10 - Arrival Procedures and Separation   

In short, there are lots of extra procedures that must be followed when using intersecting runways simultaneously, but they also get a lot of operational benefits when they can do it so are able to get a lot more airplanes on and off the ground in the same amount of time.