First I scanned in the drawing and made them really contrasty black and white images with Photoshop. This made them easier to see through the fabric. I taped it onto a piece of glass and leant it in front of the window. If I do lots of these I'll definitely make up some sort of light table (or this one) so I don't have to work on a nearly vertical surface.

Then I placed the fabric over the top and traced the image with a fabric pen. Hazel calls these her "Cherry blossom branches", although I must admit I've cobbled together a bunch of different examples here so it's not exactly an original, nor are they particularly branchy-looking! The red just seemed like the only colour choice on the white, although a nice sage green would be good too I think.

The finished tea towel - well not hemmed yet!

The rough texture of the cotton I've used gives the lines a real silk screened appearance, which doesn't come across well in the photos but it's very effective. Mat couldn't figure out how I'd done it when I showed him.

I also did a straight copy of one of Hazel's family portraits, in this case all of us out in the garden. After I took the photo I duplicated a few more of the flowers to fill the width a bit better and got her to sign it as well.

These are so quick and easy and, dare I say it, really effective! They'd make great presents for grandparents or aunts and uncles and all your many friends and relations.
Fabric notes upon further experimentation
I did a second lot of tea towels using a different, lighter fabric and have the following to say about fabric choice:
Heavy cotton used above
Pros:
It held its shape well, no stretching and
Took the fabric pen well as it had a dense weave.
Easy to draw on due to dense weave.
Cons:
Possibly too heavy for tea towels but time will tell
Harder to see through to trace - need light source behind fabric.
Lighter cotton with looser weave
Pros:
Very easy to see through, can do on flat surface with no light behind.
Probably a better tea towel fabric, softer and more flexible.
Cons:
Very stretchy, doesn't hold shape well.
Shifts under pen point
Looser weave doesn't show pen as clearly
Much harder to draw on.
The search continues for the ideal fabric on this end, but it illustrates the importance of considering what result you want and what fabric will be both a good tea towel and easy to draw on.