Hi there, fritzie-ga!
Most sites that I have visited warn against using poison, due to the
risk to other pets, as you mentioned. However, most believe that traps
are the best way to go.
Although you said you dont want to use them, this Website claims
traps are the best method for getting rid of the rodents. It also
gives some good tips for preventative measures to stop the rats from
coming back. The recommended bait is peanut butter mixed with oats and
cheese, or dried fruit.
This person had a similar problem with poison, and this site
recommends setting the traps while wearing gloves, as the smell of
humans could put the rats off the trap.
However, there are one or two other options. The use of an ultrasonic
device is another method of getting rid of rats. It emits a
high-pitched noise only the rodents can hear, so it is safe to use
with other pets around, as long as they are not rodent-type pets. It
costs about $40 and has a 60-day moneyback guarantee.
The Transonic IX-L is another of these devices; it costs £75.95.
However, it also repels birds and squirrels.
The Rodar II will get rid of what the products manufacturers describe
as pests they include squirrels in this group, but do not
specifically mention birds. This costs $89.95
However, not everyone is convinced of the effectiveness of these
devices. This Website doesnt believe that the ultrasonic devices work
in the long-term.
This discussion group came up with a couple of ways to exterminate
rats some are slightly bizarre, others are just downright cruel
(which I wouldnt recommend). One suggests using Coca Cola although
this will kill the rats.
This webiste recommends taking away their food, water and shelter,
wlthough I understand that this is not always possible.
A combination of methods might help.
I hope this helps!