Credit Card

Will my non-chip containing U.S. credit card work in NZ? Or will we have trouble using it in automated machines like the last time we went to Europe?

Technically it depends on more than just the chip and what affiliation the cards bank has. Usually if you have a PIN it is ok, but to be sure just ask your bank for a new card because you are travelling overseas(I assume all new cards issued now in the USA have a chip, like it is in Aus/NZ).
I know in Australia they are getting rid of signing for credit cards and replacing with a PIN only with chip in August.

There are still plenty using the cards without a Pin in NZ but like Australia, will at some stage be a thing of the past.

I called my credit card company and they are issuing me a new card with a chip. Thanks for the help.