I don't know where people get this idea that you have to have a chip-and-pin CC to get by in Europe. It's just not true.
I live in the UK, so examples of things you wouldn't be able to buy with a card include:
- train tickets (you'll need cash, or else a long queue if there's a human option)
- car parking (sometimes cash won't be an option, though that's rare)
- occasional smaller businesses (shops, restaurants) who will want cash instead due to the fraud risk
- any other ticket machine (e.g. cinema)
OK, it's more of an inconvenience than a necessity. It's ridiculous that the US has barely started to use the system though -- it's almost 10 years old.
(I don't think Brno is much to brag about...)
I live in the UK and almost always get train tickets using a card, both at the counter and using the machines. This is from train stations an hour or so away from London, so maybe they don't have them further away.
I have seen some smaller businesses in the village near me only accept cash, however that was mainly because there was an ATM across the road and it would cost them to process cards themselves.