Rogue Male

Geoffrey Household's Rogue Male is, according to the jacket copy, a minor classic of sorts; it was the first cat-and-mouse suspense novel, published in 1939.

Approaching it as a modern reader, there's definitely a quaintness to the novel, even aside from the classism.  The protagonist is remarkably upset when he kills one of his pursuers, early in the novel.  That squeamishness is important for stretching out the novel, as the narrator is so upset that he resolves a path of non-violence, even as the forces of a secret police are set on his trail.  But it's hard to imagine a modern protagonist being so squeamish, with his own life at stake (I think, for example, of Point of Impact or The Winter of Frankie Machine, both cat-and-mouse novels written in the last couple of decades).

Another old-fashioned bit is the relative psychological subtlety.  It takes most of the novel for the protagonist's motivations to become clear, even to himself.  Again, one can't imagine a modern thriller taking a detour through the protagonist's psyche, and I think the field is poorer for the lack.