my makeshift version is to pronounce pi, tau and kappa like the p,t and k of my ordinary English(ie with a fair bit of aspiration built in) and then for phi, theta and chi to lay on the aspiration with a trowel; the trouble is I find that puts stress on the syllable with the aspirate,
Phi, theta and chi were probably originally pronounced like English p, t and k when followed immediately by a vowel and not preceded by s; and pi, tau and kappa like French p, t and c or qu, respectively (i.e., as pure occlusives with no aspiration). Eventually phi, theta and chi came to be pronounced as fricatives, probably something like f, English th and German ch, respectively, but the evidence is not clear when these changes came about, and they probably occurred at different times in different dialects. We have some idea of the pronunciation of Attic Greek in the classical period (5th-4th centuries), but we have no idea how Ionic Greek was pronounced in the archaic period, when the Homeric poems originated.
Why not just pronounce pi, tau and kappa like English p, t and k, and pronounce phi, theta and chi like English f, English th and German ch, respectively? There are several different ways ancient Greek is pronounced, and this is one that is common in the US, at least. There are many more important things to worry about in tackling Homeric Greek than pronunciation.
No matter how hard we try, we will never know whether we're pronouncing ancient Greek properly, since there are no more ancient Greeks to correct our pronunciation. In any case, unless you're able to engage in time-travel, you'll never have a problem communicating orally with ancient Greeks. And just as almost no one can master the pronunciation of a foreign language after childhood to the point where they would be mistaken for a native speaker (even though they may master grammar and syntax thoroughly), if we had a chance to speak to an ancient Greek, we would all sound like the barbarians we are.