Hello errodr,
I think that the stepping stones will eventually "sink".
You should consider using "Dry sand-and-cement bed" under the stepping stones and have a Weed Barrier Cloth over the sand before applying the mulch on it.
However, why not consider to not use mulch but a more Japanese solution like gravel or (small) crushed stone.
Also worth considering is the use of a band to enclose the pathway. A natural (look) material like (cut) granite would be best suited.
This would also easy the lawn maintenance.
Have a look at my front entrance path (not in a lawn though ! also http://www.karesansui.nl/html/page_compfront.htm )
Regards, Piet

Here is a link that might be useful: Constructing Paths and Terraces in Tsubo-en

Errodr, Piet has good advice.
Yes, the use of sand only may result in washing out from under the stepping stones and making them unstable. The addition of cement to the sand and then sprinkling it with water serves to 'stiffen' the sand but the use of well-packed, fine gravel would be more appropriate, as would the use of that same gravel as 'mulch' between the stepping stones. I have found that placing steel landscape barrier at the grass/walk edge serves to contain the sand/gravel mix and leaves a clean line for the eye to follow. It is not traditional japanese method but it is eminently practical.

We have large round stone steps in our garden - we simply dug out the grass for each stone. They do settle into the ground a bit but look natural and are useful and beautiful.

My wife and I are doing the Asian thing completely inside and outside. We used decomposed granite as a base, then sand and then the stones. The decomposed granite will prevent wash-out.