Yes, this can work but isn't that straight forward - at least I din't find it that obvious:

Look for the agentless part in the docs for a brief rundown first:

http://www.zabbix.com/documentation/1.8/manual/installation

For agentless pings uncomment fping in Server's .conf file FpingLocation=/usr/bin/fping (get path right and install it if not). Look here: http://tinyurl.com/3lpmnwg

Choose simple checks for all services, create a host, then an item and then the trigger.

ICMP needs funny monitoring details (a multiplier etc) but check the Safari book if stuck again.

This might help a little too:

http://www.debianhelp.co.uk/zabbix.htm