no, no requirement for root position chords.
the only admonition might be to be careful when using 2nd inversion chords... they can sometimes NOT sound like the harmony they are supposed to be. (it comes from counterpoint, where a 2nd inversion chord implies a double appogiatura of the two upper notes, which must resolve down by stepwise motion).
I've written a short little musical sample, no real development, but just an idea of how the material can be treated.
I'll be posting it in the tutorial thread.
The symphony excerpt I included with the tutorial uses a dominant pedal for a lot of the passage, so you're getting that movement by 3rds but over a constant low Bb. in that case, the bass isn't really the root position, since it's this pedal tone.