How would I find out what my own specific ancestors ate? Also, what if my mother and father have vastly different lineages? For instance, my father is Persian, my mother is Russian and Polish. I have a feeling their ancestors may have eaten different things entirely.
This is a significant question.
Another way to explore this topic is to investigate genetic ancestry.
Based on your maternal and paternal line you can approximate the locations of people with your genetic ancestry at particular eras. 23andme, for example provide the location at around 500 years ago, which is before the era of significant travel. Once you know your haplogroups (groupings of genetic variants) you can investigate where your haplogroup was located in other other times.
With knowledge of the location and era you can then make a determination of the types of foods that would have been available in terms of climate, geographic features and proximity to centres of civilization.
The climate determines not only the types and seasonality of fruits and vegetables but also the amount of fat in animals. Geographic features help to refine that information as well as determining whether omega 3 source was fish-based.
This type of investigation requires some investment in time and learning but can be very rewarding.
East Europeans tend to eat a somewhat paleo, but more whole foods diet. Lots of gelatin, meats, sauerkraut, kefir, raw milk (still common in most of those countries), potatoes (white, not sweet), bland-tasting vegetables (kale/collard greens/mustard greens/arugula is not really on the East Euro menu.)
I go to a Polish butcher and they always have high quality meat and they sell gelatin with some meat and veggies in it. It's so hard finding chicken and lamb hearts anywhere except these European butchers. Even liver is hard to find in many groceries and American butchers.
Spices such as dill, parsley and peppercorns are extremely common in East Euro cooking.
As for Persian stuff, there's definitely a lot of difference. I have little experience with Persian foods, so I'll just leave that to someone else to answer.
The best thing to do is a metabolic typing test - involving a questionnaire and metabolic challenge tests. This can pinpoint your ancestral/genetic makeup and work out your diet requirements pretty accurately (giving you an idea of the percentage of protein/fat/carbs you need to be eating according to your ancestry).