Does anybody know a phonetic transcription system of Polish for non-Polish speakers?
Well there's IPA, which is international by definition, so it has all symbols necessary for a phonetic transcription of Polish.
I don't know if there are any practice books (similar to How Now Brown Cow for British English), but a good resource based on IPA that I know of is
Słownik wymowy polskiej PWN (The dictionary of Polish pronunciation). It has lots of possible variants for most words, and it includes two very useful "From letter to phone" and "From phone to letter" reference tables. There's also a dual lanugage (Polish/English) overview of the trends in Polish pronunciation and of the phonetic transcription of Polish in general.
Unfortunately there haven't been any new editions published for years now - the one I have is from 1977 - but you might try your luck in online second-hand bookshops.
By the way, Marek, there's a marked difference in pronunciation between 'trzy' and 'czy'.