Deti is pronounced dyetyi here. Is this the equivalent of a "slender" D and T, such as you would find in Irish?
If 'd, t, n' are followed by either 'ě' or 'i/í', they become palatalized.
Unfortunately, I am not familiar with Irish, so I can't answer that part of your question.
You may wish to read the introductory tips & notes for more info on Czech pronunciation.
Thank you, endless sleeper. I had a read, and, yes, it looks like the same process is at work in both languages. The tongue is on the dental ridge for a d or t followed by a, o or u, but moves to the hard palate for d, t followed by e or i, sometimes producing a y off-glide dye, tye.