É muito mais simples do que se pensa:
asd@locutus:~$ IZE=file.avi
asd@locutus:~$ echo $IZE
file.avi
asd@locutus:~$ echo ${IZE%%.???}
file
asd@locutus:~$
Então, dessa maneira:
ls -1 /a/dir/*/dir/* | while read VAR; do echo ${VAR%.???}; done
Você pode fazer o que quiser entre fazer e fazer, o eco é apenas um exemplo.
man bash (expansão de parâmetro):
${parameter%%word}
The word is expanded to produce a pattern just as in pathname expansion.
If the pattern matches a trailing portion of the expanded value of param‐
eter, then the result of the expansion is the expanded value of parameter
with the shortest matching pattern (the ‘‘%’’ case) or the longest match‐
ing pattern (the ‘‘%%’’ case) deleted. If parameter is @ or *, the pat‐
tern removal operation is applied to each positional parameter in turn,
and the expansion is the resultant list. If parameter is an array vari‐
able subscripted with @ or *, the pattern removal operation is applied to
each member of the array in turn, and the expansion is the resultant
list.