"Strøm-type" model structure on chain complexes?  Background 
The  Quillen model structure  on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by the lifting property, but in the end they are those inclusions which are built up by cell attachments (or are retracts of such things).
The  Strøm model structure  on spaces has weak equivalences = the homotopy equivalences and fibrations the Hurewicz fibrations. Cofibrations the closed inclusions which satisfy the homotopy extension property. 
The  projective model structure   on non-negatively graded chain complexes over $\Bbb Z$ has fibrations given by the degreewise surjections and weak equivalences given by the quasi-isomorphisms. Cofibrations are given by the degree-wise split inclusions such that the quotient complex is degree-wise free. 
From the above, it would appear to me that the projective model structure on chain complexes is analogous to the Quillen model structure on spaces.
 Question 
 Is there a model structure on (non-negatively graded) chain complexes over $\Bbb Z$ for which the weak equivalences are the chain homotopy equivalences? 
(Extra wish: I want cofibrations in the projective model structure to be a sub-class of the cofibrations in the model structure answering the question. Conjecturally, they should be the inclusions satisfying the chain homotopy extension property.) 
 A: Very recently a paper appeared on the arxiv by Barthel-May-Riehl which addresses this question in a very complete way. It discusses the three model structures on DG-algebras (answering the OPs question and covering the mixed model structure as well), then goes on to give six model structures on DG modules over a DGA. This paper generalizing the references in Peter May's answer above and gives model category foundations for some classical constructions in differential graded algebra.
A: There are several other papers, I think earlier ones, that cover this. 
[32] M. Cole. The homotopy category of chain complexes is a homotopy category. Preprint (1990's)
[29] J. Daniel Christensen and Mark Hovey. Quillen model structures for relative homological 
algebra. Math. Proc. Cambridge Philos. Soc., 133(2):261–293, 2002.
[121] R. Schw¨anzl and R. M. Vogt. Strong coﬁbrations and ﬁbrations in enriched categories. Arch. Math. (Basel), 79(6):449–462, 2002. 
The references are from the book ``More concise algebraic topology: localizations, 
completions, and model categories'', by Kate Ponto and myself. It will be published
this year by the University of Chicago Press.  It includes and compares three natural
model structures on spaces and chain complexes, the Strom model structure, the Quillen
model structure, and Cole's mixed model structures:
33] Michael Cole. Mixing model structures. Topology Appl., 153(7):1016–1032, 2006.
