If anyone's interested, just an update on this after the converter box and optical cable arrived. Note, I am hopeless at this sort of thing.
Firstly I needed another HDMI cable but after a spot of rummaging I found one.
Secondly the power lead supplied for the converter strangely had a USB connector not a plug on it! I have just left a phone lead in a hotel room so did have a plug with a USB input spare. Phew.
(So all set to go with high hopes...)
Thirdly I am using the DVD input on the AVR350 for the HDMI lead but then find there is no optical input for the DVD, it is a coaxial I think (which the DVD player was using successfully). There is an unused slot for Tape so plug it in there just to check. As suspected you cannot select both DVD (video) and Tape (audio) together. Damn.
So near and yet so far...time to regroup. Hmmm, how about RTFM to see if there is anything in there. Amazingly find the manual (untouched in about 10 years) and what do you know but it covers this and indicates you can reassign the Tape audio slot to apply to the DVD. So now need to access the AVR350 advanced menu...aha the old remote is in the drawer as well, lovely...batteries don't work and don't have AAA replacements, not lovely. RTFM again and find I can use the front panel instead.
Manage to change the settings and try again...nope. Two problems found, lead not plugged in firmly enough and settings not saved correctly in Setup menu (I did say I was useless).
It must work now surely...TADA!...nope.
Still lovely pictures but no sound. I'm stumped but then gazing at the display of the AVR350 I see it says Analogue on it and I'm trying to connect Digital aren't I? Rings a bell that I've had this problem before. After pressing the Direct button it switches to show Digital.
Restart everything and...it works! YeeHaw! And very nice it is too.
So after all that, yes it is possible to use Amazon Fire TV with an AVR350 by utilising the DVD slots (so can't have DVD set up too) and a signal convertor.
Thanks for the advice.