I don't think they're remotely old fashioned! They're on trend yet fairly classical, imho. For inspiration, look at Sophie Robinson's kitchen...although her floral walls are wallpaper.
Definitely don't break them up by adding plain tiles as that would look dated.
I'd also agree to go with a punchy colour on your walls or cabinets as suggested by @Ketzele.
We haven't used tiles in a kitchen since our last-but-two house in 2012, but if I was going to (and coincidentally we did consider floral tiles - but much larger format - in our current house where we're in the process of doing the kitchen), I'd definitely choose something I loved rather than playing safe with a neutral.