There's a section in the below link that describes a cable modification that may work in your situtaion (it's about 75% of the entire page down -- also the site is black background w/white text).
http://www.algonet.se/~dennisgr/z88-dark.htmIt begins:
DOS by itself does NOT use Xon/Xoff and it does not use hardware flow control when receiving but it does when sending.
The Z88 hardware flow control is always active so when trying to send, the Z88 never receives a positive voltage on pin 5 CTS to be able to send.
One useful source for a positive voltage is the PC pin 20 DTR instead of the PC-Link connection to pin 4 RTS. The connections would then be exactly the ones recommended in the Z88 User Manual for connecting to a printer, apart from the fact that the printer needs a male connector. A printer cable and a gender changer will do the trick.
No flow control is then active when sending to the PC. I only guess it is OK as long as the received data fit in the available RAM and must not be written to disk and interrupting the possibility to correctly receive data.