Good day, Headcase. You must have played that flute just across the city from where I am typing now.
Flat picking is just the same as fingerpicking - it takes practice and patience. Beyond that essential, there are a few wrinkles that may help.
1. Get yourself a medium to heavy pick and stick with it. Don't go for a thin and whispy one as when you transition from chords to single notes you won't have the grunt to make them ring out. I find that Dunlop tortex yellows are thin, but the material has a rigidity that is perfect for my playing - but its a personal decision.
2. When you are playing chords hold the pick loosely. I find that moving my index finger higher up the pick helps the pick to develop that crucial waggle (?!) and keep the pick in your mit.
3. When picking single note runs, the angle of attack that you give to the pick will also make a difference. There are two dimensions to this. Firstly, hitting the string flat on will produce a ringing note, hitting it at 45 degrees will soften the note. Secondly, twisting the pick so that more of the edge of the pick comes into contact will actually make it easier to pluck the string. I learnt through teaching myself to play mandolin ( which has double the string courses and double the plucking problems ) that twisting the pick increases the speed you can pick while maintaining the volume.
4. Try some cross-picking exercises to develop control of the pick in relation to the strings. Folk of the wood have free lessons at http://www.folkofthewood.com/page5308.htm .
Again, for more in depth discussion about picking check out Folk of the Wood . http://www.folkofthewood.com/page3592.htm.
