I believe the next three Robicheaux novels are Heaven's Prisoners, A
Morning for Flamingos and Black Cherry Blues. But am not sure of the
remainder. My mission here, is very simple. I prefer to read in order.
Will you help?

Here is a list of James Lee Burke's Robicheaux books, in order:

1.  The Neon Rain (1987)
2.  Heaven's Prisoners (1988)
3.  Black Cherry Blues (1989)
4.  A Morning for Flamingos (1990)
5.  A Stained White Radiance (1992)
6.  In the Electric Mist with Confederate Dead (1993)
7.  Dixie City Jam (1994)
8.  Burning Angel (1995)
9.  Cadillac Jukebox (1995)
10. Sunset Limited (1998)
11. Purple Cane Road (2000)
12. Jolie Blon's Bounce (2002)
13. Last Car to Elysian Fields (2003)
14. Crusader's Cross (2005)
15. Pegasus Descending (2006)

Fantastic Fiction
http://www.fantasticfiction.co.uk/b/james-lee-burke/

The same titles, in the same order, appear on this list by an Amazon
user, with links to the Amazon listing of each book:

Amazon: James Lee Burke's "Dave Robicheaux" Books in Order 
http://www.amazon.com/gp/richpub/listmania/fullview/R36H4AJYVZGPNV/

My Google search strategy:

Google Web Search: "james lee burke" robicheaux "in order" OR series
://www.google.com/search?hl=en&q=%22james+lee+burke%22+robicheaux+%22in+order%22+OR+series

I hope this is precisely what you need! If anything is unclear or
incomplete, or if a link doesn't work for you, please request
clarification; I'll gladly offer further assistance before you rate my
answer.

Best regards,
pinkfreud