Although the OP pointed to Plato and the notion of subconscious in particular, it may be interesting to know that Freudian theory has some interesting precursors in hellenistic philosophy - to which Freud himself refers in his work, concerning his theory of dreams.
In his The Interpretation of Dreams (1913, Ch. 3) Freud writes:
I am far from wishing to assert that no previous writer has ever thought of tracing a dream to a wish. […] Those who put store by such hints will find that even in antiquity the physician Herophilos, who lived under the First Ptolemy, distinguished between three kinds of dreams. […]
After having presented the three different kinds, Freud continues:
The unscientific world, therefore, has always endeavoured to interpret dreams, and by applying one or the other of two essentially different methods. The first […] is symbolic dream-interpretation […]. The second […] might be described as the cipher method. […] An interesting variant of this cipher procedure […] is presented in the work on dream-interpretation by Artemidoros of Daldis.
The worthlessness of both these popular methods of interpretation does not admit of discussion […]. So that one might be tempted to grant the contention of the philosophers and psychiatrists, and to dismiss the problem of dream-interpretation as altogether fanciful.
I have, however, come to think differently. I have been forced to perceive that here, once more, we have one of those not infrequent cases where an ancient and stubbornly retained popular belief seems to have come nearer to the truth of the matter than the opinion of modern science.
In a previous footnote in Ch. 2, he writes:
Artemidoros of Daldis […] left us the fullest and most careful work of the Greco-Roman world on the interpretation of dreams.
Note: The connection, as well as the quotations, can be found in Ch. 7.3 "The Theory of Dreams" of Lucio Russo, The Forgotten Revolution, Springer 2004, p. 214-218. (A great if controversial book)
Additional discussion of and material about the link Artemidoros-Freud can be found in:
Price, "The future of dreams: From Freud to Artemidoros", in: Before Sexuality: The Contruction of Erotic Experience in the Ancient Greek World, CUP 1990.
Chapter 12: "Artemidoros, Interpretation of Dreams", in: The Interpretation of Dreams & Portents in Antiquity, Bolchazy-Carducci 1976, pp. 53-74.