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.
As Raymond Chandler is an American one must suppose that he equates 'chintzy' with cheapness, and 'she spoke with a cheap voice'- presumably less than eloquent, with a working-class idiomatic way of speaking. It does sound highly patronising, however.
In Britain I might describe someone as having a working-class accent, since there clearly is such a thing. But disparaging such a way of speaking is, if I might say so, 'a bit chintzy'.