When you write in the ancient world, all the standard forensic devices of the last 100 years disappear. No DNA, no fingerprints, no microscopes, no clever chemical analysis.
This is incredibly liberating. Mysteries are puzzle books at their core. It’s wonderful for me to be able to concentrate on the puzzle without having to worry if my byzantine plot might have been solved in the first ten pages with some obvious piece of forensics. Far from enhancing modern mysteries, forensics has put a tight envelope around what the author can do without looking silly. It’s a particular problem for cozies. Cozy authors must sometimes go to ridiculous lengths, or fudge outrageously, to find a reason why the Chocolate Cake Killer couldn’t have been found before the end of chapter one with a quick sweep for DNA (saliva on the cake crumbs...).
A particular trap is to translate back into ancient times, techniques which are not outright impossible, but which assume a diagnostic practice far in advance of known science. It's not impossible for someone in Classical Athens to take a plaster cast of the killer's footprint, to match it with the suspects' sandals in search of the one with the diamond shaped pebble lodged in the right heel, but seriously, this is a stretch, and it tries to turn ancient investigations into a poor man's copy of modern ones.
A real-life Nicolaos would tread in the footprint beside the body, ignoring the killer's blood drops and those smudgy marks on the handle of the knife and the hairs caught between the victim's fingers during the struggle, as he walks to the nearest well to see if someone had dropped in a curse tablet. Because a curse tablet might be important evidence. In his defence, the real-life Nico would note from the footprints that there was a single killer, that he used a knife, and that there was a struggle.
Conversely the ancient detective has opportunities that a modern detective could never imagine. In an age when everything is hand-crafted, you can pick up any item and trace it back to the artisan who made it, if you're persistent enough. Indeed to this day an expert on ancient pottery can tell you in which city a particular piece was made. Believe it or not, if it's from Athens, they can sometimes narrow it down to which workshop made the pot!
A bowyer can examine a bow and tell you everything there is to know about its flight characteristics. An armorer can tell from the binding if a spear was made by a left or right handed man. Try doing that with mass production.
To cut the dead is deeply sacrilegious, so no autopsies, but any man could glance at a fatal wound to tell you what sort of blade made it. Every Athenian man has served in an army that used nothing but edged weapons. Everyone has seen plenty of corpses and know what happens to them over time. For the same reason, blood spatter patterns are all too familiar.
Few poisons are known, but those few are known well, and a trip to the local pharmakis will tell you everything you need. Every city mints its own coins, and the designs changed over time. You can tell more from a dropped ancient coin than a modern.
There's an extremely high reliance placed on eyewitness testimony, which is valued above all else in court, while physical evidence is looked upon as a bit dodgy -- the exact opposite of a modern court.
Clothing runs to a standard form but the decoration is totally individual. There's no such thing as identical T-shirts or anonymous suits. Sandal and clothing decoration had styles which can be spotted.
A sculptor can glance at a piece of marble and tell you which quarry it came from. Painters make their own paints. A silver armband can be identified as Phoenician from the imagery alone. The body with the blue tattoos across half his face is probably not from Athens.
I try to play to the strengths of the period. Don't try to turn ancient forensics into faux-modern. Instead, try very hard to see it the same way someone of the times would, and deliberately go for the evidence which is most different from modern expectation. It's a great way to give readers a tour of the ancient world while sticking tight to the plot line.