There are lots of plants that would flower in part shade (astilbe, bleeding heart, violets, mountain laurel, hosta, speedwells, columbine) but I can't think of many that'd be good cut flowers.

Is the shade cast by a tree, or by a building? If it's a deciduous tree, you could plant daffodils for cutting. They'd bloom before the tree leaves out fully.

I have dahlias in a window box that gets about 4 hours of morning and midafternoon sun. They seem to be doing very well. Hydrangea should also do well, oakleaf hydrangea is a nice native, though there is a new hydrangea which bloom repeatedly on new wood. (All other hydrangea's bloom on old wood, which is why you don't want to cut back a hydrangea.) You might also try Globe Amaranth, Foxglove, and Butterfly Bush (traditionally a full sun plant, but mine blooms in a partially shaded area).

There is another hydrangea that does well in shade or sun and flowers on new wood. Try looking for Annabelle hydrangea.
It can be a little floppy but only because it flowers so much.... and the white flowers really light up a partially shaded area. I think you'll like it.
how about hostas? Some have nice flowers as well as leaves that can be cut. 'Royal standard' has pure white blooms that are fragrant too. I like this one more for it's blooms than it's foliage, plus it's indestructible (unless you have deer and voles). Hey I just learned that this was the first patented hosta back in 1965!
You might have luck with a few lilies. 4 hours might be enough sun. Other than that I'm out of ideas.

Here is a link that might be useful: annabelle

Jim:
In part it depends on whether this is a dry or a moist area.

Some that work for me and are good for cutting: Columbines, Feverfew, German Chamomile, Black Eyed Susan.

You might experiment with flowers that are for both sun and shade.

Anita