Cash based commerce should just be rounded to the 1/10th dollar not the 1/100th dollar; there's just no need for such precision.
I'd just announce that while still legal tender, the penny, nickle, and quarter won't be minted any longer except in low volumes for collectors, and that cash transactions may be rounded down to the nearest 10th dollar if proper change is not available.
IE a cup of coffee costing $4.17 would cost $4.10 if paying cash. I suspect vendors would just update the cost of everything to land on the 1/10th increment.
I guess you may need to worry about people playing arbitrage games with metered things like gasoline, but even there it's probably just a marginal problem.
Dime is a weird word; it doesn't convey that it's a tenth of a dollar, does it?
[edited to add examples and digression on gasoline]