What does "Chintzy Voice" mean?

I don't think the American sense of cheap or tacky is meant here. Chandler was educated at Dulwich College and didn't return to America until he was in his twenties. So there were strains of British culture in his background, and I think his use of the word chintzy is more in line with that. People who favoured chintz, with its bright floral patterns, were people who might be working class in origin, but were also people with social aspirations. Their front rooms were likely to be over-decorated, stuffed with knick-knacks, and with aspidistras in the window (like the Comstock's home in Keep the Aspidistra Flying). George Orwell's word for that class of people was "shabby-genteel". What I think Chandler was trying to portray was someone from that kind of background, rather prim and proper, someone who puts on airs and graces and is desperate to be seen as "respectable". I haven't read the book (or, if I have, I don't recognize the quotation), so I could be wrong. But that's the way I read it.

The word 'chintz' refers to a multicoloured cotton fabric, with a glazed finish. The word is Hindi in origin and dates from the early 17th century. The Hindi word 'chint' means a 'spattering' or 'stain'.

When it comes to 'chintzy', British and American English diverge. In Britain it is still connected to fabric. 'A pretty chintzy pattern'.

In North America it connotes cheapness and poor quality.

Having read @Terpsichore's answer I have significantly edited this one, having earlier taken Raymond Chandler to have been an American. However he would appear to have spent his young life in Britain, having been educated at Dulwich college. And this fact suggests his usage here was more a British one. So I accept Terpsichore's view and have upvoted that answer.