In France, Pierre is pronounced / p j ɛː ʁ / , like the common noun which means stone. You can hear it here. I however switch to / p j ɛː ɻ / in English (like here) after several failed attemps at making myself understood.
In Croatia, my name is spelled Dragičević and pronounced / d r a g i ʧ ɛː v i ʨ / (similar to the English pronunciation explained before), with a stress on the second and third syllables. You can hear it at the beginning of this recording.
To make it easier for French people, my father removed the diacritics and gallicized its pronunciation into / d ʁ a ʒ i s e v i k /, which significantly differs from the original one. Interestingly, this makes things much harder for English speakers.
The inverted circumflex on the č is called a caron. Phonetically, the distinction between č and ć is somewhat subtle.
IPA. Handbook of the International Phonetic Association : A Guide to the Use of the International Phonetic Alphabet. Cambridge University Press, June 28, 1999. ISBN: 0521637511.
This book is an excellent introduction to phonetics. Contrary to a common belief, phonetics is not boring. Getting acquainted with the logics of phonetics will make your verbal automatisms crystal clear, which is very rewarding both intellectually and pragmatically.
The University of Victoria also has a nice phonetic chart with audio recordings.