I thought about it and tried to think about what I do as far as improvising with the right hand goes. Basically there are really only a few things you can do. You can:

1. Play single note runs - this is where you just play one note at a time

2. Play arpeggios - broken chords up and down the keyboard

3. Play chords - using 3rds or sixths or any combination of the chord-scale relationship

Different genres of music use the above techniques in different ways. New Age piano usually relies on a softer sound although there is a "minimalist" style that uses hard sounding chords in the right hand ala George Winston.

You see, the problem is that you have to know the numerical relationship between the root note and the rest of the notes that make up the chord to "know what you're doing" on the keyboard.

A book I highly recommend, although it may not be on the market anymore is Alan Swain's Four-way Keyboard system. It's not an easy instruction book, but it is one of the best for how chords are constructed - thus allowing you to know when you are playing 3rds or 4ths and so on. Not necessary for making music, but nice to know anyway.