Can we predict with any reliability whether a chemical is carcinogenic in people?
One of the great problems is that there is no single mechanism by which a chemical carcinogen causes the damage which leads to cancer. There are a range of different mechanisms, I will mention some of them.

One mechanism is alkylation of DNA, the attachment of alkyl groups to DNA will prevent it being read correctly. In some cases the damage can be repared while in other cased the damage will cause the body to misread one base pair as another. There is an interesting problem with electrophilic alkylation agents, if they are too reactive towards water then they will not survive to reach the DNA.

There is an interesting paper in which this idea is discussed E.W. Vogel and M.J.M. Nivard, Mutation Research, 1994, volume 305, pages 13-32 considers it at length. If we consider the ratio of the rate constant for the reaction of an alkylation agent with thiosulfate and acetate, then as the ratio of these reaction rates changes in favour of the thiosulfate reaction something known as the Swain Scott constant (s) also increases. The Swain Scott constant is the ratio between reaction at two different sites (7-alkylguanine/O6-alkylguanine) in DNA. In general the higher the value of the s constant the stronger a carcinogen is. In the paper by A. Barbin and H. Bartsch in Mutation Research, 1989, volume 215, page 95 to 106 a series of directly alkylating carcinogens a good but not perfect relatioship is seen between the s constant and how carconogenic they are in rodents.

So it appears for one class of carcinogen that it is possible by doing some simple chemical kinetics experiments that it is possible to make an assessment of how carcinogenic a substance.

One of the great problems is that many "carcinogens" are in fact "precarcinogens" which require metabolic activation to make them carcinogenic. For example hydrazine reacts with formaldehyde in the liver before being oxidized to form diazomethane which then methylates the liver. This is why acute large hydrazine exposures in humans can cause liver failure.

This brings us to the Ames test, this is a test using bacteria which measures how able a substance is to alter DNA. The idea is that cancer is caused by altering the DNA of normal cells. So something which is able to randomly alter the DNA of cells is able to induce cancer. The great problem with the Ames test is that while it is a good test. There are both false positives and false negatives if it is used as a test for carcinogenic potential.

The normal Ames test protocol includes some sterile extract of rat liver to provide some metabolic activation. This will increase the ability of things like polycyclic aromatic hydrocarbons to alter the DNA of the bacteria.

If we consider an inorganic example, the chromate anion is known to be carcinogenic while chromium(III) is not carcinogenic. To understand this we need to know something about chromium biochemistry.

It is true that chromium(III) can form some very stable complexes which are very slow to react with DNA. But the chromium(III) is unable to reach the DNA in the cell. However chromate is able to pass through the sulfate transport channels in both the cell and nuclear membranes to allow it to reach the nucloplasm. There it is reduced to chromium(III) which then damages the DNA. As a result we can make a judegment about chromate and sulfate.

Sulfate is unable to react in the nucloplasm to form something which will form long lasting complexes with DNA, so even while sulfate is able to reach the DNA by the same route as chromate it is clear that sulfate is unable to act as a carcinogen by the same mechanism as chromate.

Also permanganate or magnanate is a stronger oxidant than chromate, I suspect that as it would be less able to survive the journey through the body to the DNA it is less able to incude cancer. This is likely to be one of the the reasons why permanganate is far less carcinogenic than chromate. Plenty of evidence exists for chromate being carcinogenic but little if any evidence exists for permanaganate. 

Also the complexes of manganese with DNA are likely to be shorter lived than those of chromium. This is becuase chromium(III) complexes are slow to exchange ligands while manganese(II) are rather quick to exchange ligands. This is due to the fact that chromium(III) complexes are d3 while manganese(II) are d4.