I have harvested seeds from Zinnia elegans and Dahlia variabilis and I don't know how to easily separate the seeds from the chaff. I have put them in a bag and shaked it hoping that the seeds will fall at the bottom, but only a few did.
What else can I try?
Personally I would start by picking out the large debris with my hand. From there, I've seen people take the seeds and place them in a shallow dish, tilt the dish, and gentle blow the chaff away - this is called winnowing. There are also commercial seed sives that can the separating for you purchase, but I believe they are expensive. The are usually a series of mesh screens with openings decreasing in size.
I'll post this as an answer because it is more of an answer than an update.
The easiest way of separating the seeds was rubbing them when dry in order to grind the chaff (as Bamboo says in her comment), then winnowing (as in Ben's answer that I will mark as accepted). For more round-ish seeds it worked well by improvising a sive from window screen (as in Wayfaring Stranger's comment).
The result looks like this:
