When I was studying (classical) Greek as an undergrad, we were told that φ θ χ, that is, the Greek aspirate consonants, have the following phonetic values: [f θ χ] and never it was told to us that they should actually be pronounced as [pʰ tʰ kʰ], respectively, which correspond the their phonemic forms /pʰ tʰ kʰ/. What really puts me off is the fact that all complex, idiosyncratic phonological rules that characterize Greek are easier to explain when we use the true forms instead of the other ones. Nevertheless, nobody ever spoke about the true forms. Does anybody know how to solve this (potentially historic) conundrum?
Well this sound change took place at different times in different dialects/regions.
Some claim that fricativisation already started in the late 6th/early 5th century BC, but that is controversial (see my reference below). In Laconia there is definite evidence for θ from the 4th century BC where ἀνέσηκε is attested in inscriptions instead of ἀνέθηκε and σιός instead of θεός. By the 1st century AD this sound change had spread everywhere. If you are interested in the chronology of the Greek sound changes as a whole, this sums it up nicely: "The Greek language and its speakers".
A fricative pronunciation of /pʰ/ is attested in Pamph. phíkati = wíkati ‘twenty’ (4th c. BCE). Spirantization of /tʰ/ also occurs quite early in Laconian: siós = theós ‘god’ (Laconia, 4th c. BCE; cf. Tsakoniansáti ‘daughter’ < Anc. Gk. thugátēr). Other evidence is controversial. p.n. Ephronís for Euphronís (Athens, Hell.) might reflect an assimilation of contiguous homorganic fricatives (/ɸpɸ/ > /ɸɸ/ with simplification to /ɸ/ before a consonant), but careless omission of <u> is more likely. The spelling of <ph> for <th> in worphaía = worthaía ‘(Artemis) Ortheia’ (Laconia, 6th c. BCE), pheôn = theôn ‘god (gen. pl.)’ (Dodona, Hell.) purportedly reflects spirantization of /tʰ/ and /pʰ/, since the fronting /θ/ > /f/ is more natural than /tʰ/ > /pʰ/ (Cockney three [friː]). But a confusion between <Θ> = th and <Φ> = ph may also be behind these examples (Threatte 1980:470-471). However, in Pompeian graffiti (1st c. CE) <f> for <ph> probably mirrors a fricative in the Greek words: Venus Fisica. Arguably, voiceless stops underwent spirantization and voicing in Macedonian: p.n. Boulomága /βuːlomáɣaː/ < /Φuːlomáxaː/ for Phulomákhē, month name Xandikós /ksanðikós/ < /ksanθikós/ for Xanthikós (Hatzopoulos 2007, Méndez Dosuna 2012). Furthermore, these spellings confirm the spirantization of voiced stops in the area. Spellings like gégrappha = gégrapha ‘I have written’ (Caria, 2nd c. BCE), metēllakkhótas = metēllakhótas ‘change (pf. ptc. act. acc. pl. masc.)’ (Caria, 2nd-1st c. BCE) are unlikely to attest to an affricate pronunciation prior to complete fricativization. Lack of spirantization of /tʰ/ and /kʰ/ after /s/ is confirmed by early spellings with <t> and <k> in NW Greece and elsewhere: heléstai = helésthai ‘seize (aor. inf. mid.)’ (Chaleion, 5th c. BCE), páskoi = páskhoi ‘suffer (3 sg. opt. pres.)’ (Olympia, 5th c. BCE). Apparently, /tʰ/ spirantized after /s/ in some areas (Méndez Dosuna 1985:364-366): apodóssai = apodósthai ‘give back (aor. inf. mid.)’ (Olympia, 4th c. BCE). The scantiness of <sp> for <sph>, e.g. aspalísai = asphalísai ‘secure (aor. inf.)’ (Egyptian papyrus, 1st c. CE), also indicates spirantization of /pʰ/ after /s/, which is confirmed by Modern Greek.
Alcorac Alonso Déniz, “Spirantization”, in: Encyclopedia of Ancient Greek Language and Linguistics, Managing Editors Online Edition: First Last. Consulted online on 04 February 2017 http://dx.doi.org/10.1163/2214-448X_eagll_SIM_00000536 First published online: 2013
It is generally accepted nowadays that φ θ χ were pronounced as aspirates in Classical Attic. But this pronunciation didn't last long: as early as Aristophanes θ had become a fricative in some dialects (his stereotypical Spartan characters frequently substituted σ for θ, which was probably an attempt to transcribe a dental fricative using the closest available sound). And in some graffiti from Pompeii, φ was transcribed into Latin with f rather than ph.
By the first few centuries CE, the transition to fricatives was complete, and has remained in Greek ever since. So until fairly recently, most scholars used the same sounds when talking about Ancient Greek.
We know about the historical aspirates now through a variety of means. But since English distinguishes fricatives and does not distinguish aspirates, Greek learners in America still tend to use this "Erasmian pronunciation". Similarly ε/ει/η, ο/ου/ω are seldom given their reconstructed Attic sound values.
The spriantalisation of the Greek aspirated stops took place at different times in different dialects. “The telltale transcription of φ by Latin f instead of ph is not found until the 1st century AD, and is not usual until the 4th” (Sihler p. 142).