Now arsenic has been mentioned and interestingly, inorganic arsenic compounds interfere with the basic mechanism of DNA repair, which apparently increases frequency of chromosomal aberrations. So this is chemically predictable.
Cadmium has also been mentioned, but what is little known is "even tumor formation  can be suppressed by the simultaneous Zn treatment" cited in this work. So chemically predictably treated.
For organic compounds, I suggest a simple process to determine the POTENTIAL carcinogenic ability of an organic compound (I believe this is what you are, in effect, asking in being able to 'tell').
In the lab or reported in the research literature, is it apparent that the compound readily undergoes decomposition creating (directly or otherwise) radicals or groups in an environment of aqueous saltwater containing transition metals, CO2, oxygen and naturally produced HOCl or other reactive oxygen species (ROS) at pH levels found in the human body?
Some generated radicals, like the hydroxyl radical and the carbonate radical, for example, can attack DNA as it would any large organic compound to extract a hydrogen atom.
Damaged DNA can lead to cancer cells.
While, for example, short-lived hydroxyl radicals are good in breaking down problematic pesticides in natural waters, they are not recommended in large amounts in the human body.
Note, I am not recommending the direct testing of the decomposition of DNA, but just the expected (and conveniently measured) side products created with the associated breakdown of the organic compound in question. This may be suggestive of potential DNA damage, and at best, a predictive tool. Varying dosing levels may also be quite informative. This is categorically distinct from expensive short-term animal studies (examining tumors) and also investigations of long term low dose exposure and reported incidents of cancer in human populations with inferences drawn from correlation analysis.
If a compound displays a chemical potential to be carcinogenic, per above, you may still want to await the opinion of say Germany, England, or other countries with socialized medicine,.., to verify. But, in my opinion, I would just avoid/limit exposure and consume over the counter anti-oxidants (Vitamin C,...), especially if chemically related suspect compounds are now publicly believed to be carcinogenic.
Thanks Andrew for the reference to the Ames test! It, however, apparently employs "several strains of bacteria (Salmonella, E.coli) that carry mutation" which could limit the accessibility of the test.
DISCLOSURE: I have no financial interest, or am I employed, in the health supplement industry or the health care industry.