A useful model for topics like this comes from McCrae and Costa (1999):
There's quite a lot more going on here than is pertinent to your question, but note that influences originate from biological bases on the top left, and from external influences on the upper right. Everything else is modeled as an effect of mediated, dynamic processes between these two...except basic tendencies. McCrae and Costa's theory claims that "basic tendencies" such as the Big Five are in fact those aspects of innate personality you'd probably want to test: truly innate, and separable from external influences. However, McCrae and Costa (2003) themselves had to back off of this claim somewhat and present a more qualified model, so the goal remains elusive by practically all accounts.
Behavioral genetics is making a lot of progress in sorting out what's definitionally innate, but there's a lot of progress left to make. Half of the progress is in discovering the complexities that explain why simpler theories don't work as often as we'd like. Dealing with those complexities is another matter in itself. I agree that tests of genetic factors that influence behavior are very valuable, and far too few at present, but we're definitely working on it.
Another approach that's making a lot of progress on the experiential side is cross-cultural psychology. The story is much the same as with behavioral genetics: our access to the methods for investigating these influences is growing rapidly, hence so are our understandings of both the questions and answers. Ask again in another decade and we'll have much more to tell you than we do right now.
@Ana raises a great point that complicates matters quite a bit: gene-environment interaction. So much of who we're born as determines how we interact with our parents, and whether we choose to be like them or different. Choices of identity thus reflect both our genetic heritage and our environmental interactions. Genetic influences even affect whether we experience given stimuli as positive or negative, and then go on to influence how we choose to react. Over time, it becomes very difficult to separate environmental influences from innate influences, because our choices shape who we are, and our identities become innate in many ways too. Another useful term in this vein of theory is reciprocal determinism, which refers to this sort of bidirectional causality we see between innate and environmental factors in development.
In my limited experience teaching on these topics, I've found some particular examples of reciprocal determinism especially illustrative of both the complexity of the problem and the degree of progress we've made. Two worth offering here are as follows:
- Monoamine oxidase A (MAO-A) is sensitive to genetic variation that decreases its concentration in the body. MAO-A deficiency increases stress vulnerability (or diathesis: see another answer of mine for another example of the diathesis-stress model) in terms of the likelihood of aggressive maladaptation (Brunner, Nelen, Breakefield, Ropers, & van Oost, 1993). In children born with the gene that leads to MAO-A deficiency, a stressful environment such as one involving child abuse will be very likely to cause antisocial behavior in those children later in their lives, whereas children without MAO-A deficiencies tend to withstand such stressful environments better (Caspi et al., 2002). Furthermore, aggressive tendencies are much more similar regardless of MAO-A levels when childhood environments aren't so stressful. Thus one could test for the relevant gene to identify an innate vulnerability to aggressive maladaptation, but whether a person with genetic risk would see that risk realized would remain to be determined by the environment.
- Genetic variation may also determine the environment at a cultural level to some extent. This is where the worlds of behavioral genetics and cross-cultural psychology play very well together. An example of why we really need tons of research in both and cross-pollination between their respective theories arises in the "A118G single nucleotide polymorphism of the mu-opioid receptor gene (OPRM1)," which, first of all, relates to pain sensitivity (Fillingim et al., 2005). The global cultural relevance emerges clearly in this figure (Way & Lieberman, 2010):
Bear in mind that this is correlational evidence though; hence it's also worth considering whether cultures in these areas could've selected certain alleles like A118G over the course of history. A great many factors powerfully influence cultures besides genes, after all (Diamond, 1998; see for a book review Venkatachalam, 2001).
The 5-HTTLPR gene may relate to innate tendencies toward anxiety, but doesn't seem to relate consistently (Ebstein, 2006). Culture may have something to do with this (Chiao & Blizinsky, 2009):
- Top left: collectivism (vs. individualism), from low in yellow to high in red
- Top right: frequency of the anxiety-prone allele of 5-HTTLPR
- Bottom left and right: prevalence of anxiety and mood disorders, respectively
These maps demonstrate that despite the much higher genetic vulnerability to anxiety in China vs. USA (top right), anxiety and mood disorders appear to have much lower prevalences there than here (speaking as an American myself; see the bottom two maps). Chiao and Blizinsky argue that the higher genetic vulnerability to anxiety in China might've led to a collectivist culture there through a coevolutionary process of cultural adaptation to an ethnically linked genetic predisposition. It's an interesting argument that these innate factors could actually lead to such a strong compensatory shift in culture that the innate influence would be not only dwarfed, but reversed. As to whether that's a falsifiable theory, on the other hand...You'd probably want to read and judge for yourself; I get a little fuzzy on the details here. I suppose I make a decent example of how hard it is to keep up with theoretical advances like these though.
Aside from all that, understanding how personality develops is tricky enough already without trying to separate genetic from environmental factors. There's a rich history of debate on the degree to which stable personalities even exist, and the general consensus now is that they do, but reaching this conclusion took a lot of work (see the person–situation debate), much of which is far from settled in the details. Personality does change over time, but subtly, and with resistance – a useful simile is that personality sets like plaster over time (Srivastava, John, Gosling, & Potter, 2003).
- Brunner, H. G., Nelen, M., Breakefield, X. O., Ropers, H. H., & Van Oost, B. A. (1993). Abnormal behavior associated with a point mutation in the structural gene for monoamine oxidase A. Science, 262(5133), 578–580. Retrieved from http://www.bioforensics.com/conference07/DNA_and_behavior/Abnormal_behavior.pdf.
- Caspi, A., McClay, J., Moffitt, T. E., Mill, J., Martin, J., Craig, I. W., Taylor, A., & Poulton, R. (2002). Role of genotype in the cycle of violence in maltreated children. Science, 297(5582), 851–854. Retrieved from http://www.crassh.cam.ac.uk/uploads/documents/Caspi1.pdf.
- Chiao, J. Y., & Blizinsky, K. D. (2010). Culture–gene coevolution of individualism–collectivism and the serotonin transporter gene. Proceedings of the Royal Society B: Biological Sciences, 277(1681), 529–537. Retrieved from http://rspb.royalsocietypublishing.org/content/277/1681/529.full.html.
- Diamond, J. M. (1998). Guns, germs and steel: A short history of everybody for the last 13,000 years. Random House.
- Ebstein, R. P. (2006). The molecular genetic architecture of human personality: beyond self-report questionnaires. Molecular Psychiatry, 11(5): 427–445. Retrieved from www.nature.com/mp/journal/v11/n5/full/4001814a.html.
- Fillingim, R. B., Kaplan, L., Staud, R., Ness, T. J., Glover, T. L., Campbell, C. M., ... & Wallace, M. R. (2005). The A118G single nucleotide polymorphism of the μ-opioid receptor gene (OPRM1) is associated with pressure pain sensitivity in humans. The Journal of Pain, 6(3), 159–167.
- McCrae, R. R., & Costa Jr, P. T. (1999). A five-factor theory of personality. Handbook of Personality: Theory and Research, 2, 139–153.
- McCrae, R. R., & Costa, P. T. (2003). Personality in adulthood: A five-factor theory perspective. Guilford Press.
- Srivastava, S., John, O. P., Gosling, S. D., & Potter, J. (2003). Development of personality in early and middle adulthood: set like plaster or persistent change? Journal of Personality and Social Psychology, 84(5), 1041–1053. Retrieved from http://j.b5z.net/i/u/2043019/i/PersonalityDevelopmentInAdults.pdf.
- Venkatachalam, S. (2001). Guns, germs and steel: A short history of everybody for the last 13,000 years. Resonance, 84–88. Retrieved from http://www.ias.ac.in/resonance/Volumes/06/01/0084-0088.pdf.
- Way, B. M., & Lieberman, M. D. (2010). Is there a genetic contribution to cultural differences? Collectivism, individualism and genetic markers of social sensitivity. Social Cognitive and Affective Neuroscience, 5(2–3), 203–211. Retrieved from http://www.replicatedtypo.com/science/genetic-components-and-cultural-differences-the-social-sensitivity-hypothesis/1620/.