As the question in the title states I am wondering what material can be effectively used to shield gamma rays apart from lead? I believe concrete is often used, but it is nowhere near as effective as lead (6 cm to match 1 cm of lead as I understand it). I also hear significant bodies of water helps, as does tightly packed dirt, but surely there must be other materials that shield nearly as effectively as lead?
There are three processes by which gamma rays interact with matter: the photoelectric effect, Compton scattering, and pair production.
The photoelectric effect is an interaction between the gamma ray and an electron. It's forbidden by conservation of energy and momentum unless there is some other body present as well, such as an atomic nucleus. For this reason, the probability of the photoelectric effect is proportional not just to the density of electrons but also approximately to $Z^n$, where $Z$ is the atomic number and $n$ is about 4 to 5.
Compton scattering can occur without the presence of anything besides an electron, so it only depends on electron density.
Pair production goes like $Z^2$ at typical gamma-ray energies.
For these reasons, the best shielding against gamma rays is achieved with a substance that has a high density of electrons (which correlates with a high mass density) and also a high $Z$. Lead has these properties. It's also cheap. There are elements with a higher $Z$, such as bismuth, polonium, and uranium, but they aren't cheap, and their atomic numbers are only slightly higher.
There is nothing magical about lead for this purpose. The driving factor is the number of electrons per unit volume, which reduces (to a first approximation) to the mass density.
You get very good (better than lead) shielding performance from gold, tungsten, mercury, etc; and quite reasonable performance from iron or copper.
Question for the student: why is lead such a common choice for this application?
Lead is used commonly for two reasons. Being heavy metal it occupies less space; cheaper than other metals like Gold; and significantly absorbs low and medium energy gamma radiation.
Materials like concrete can also be used but occupies much space because thick concrete is needed. Detailed data is available on the Internet and in books on Radiological Physics and Radiation Protection. However, Research on newer shielding materials is very much needed.