First, all religions do not teach peace. It is a mischaracterization to state that they do and unfortunately, perpetuates the tendency to view all religions as basically the same. For example, Islam is derived from the Arabic word meaning submission. Not only are Muslims to submit to Allah, they are to subdue (sometimes violently) others until all are under the rule of Islam.
Second, a sufficient worldview will provide an explanation for why things are the way they are. For example, Christians believe that the world was created good, peaceful. However, due to free will, humans chose to act against the created order, thus disturbing created peace. Christians call this interruption in the created order "sin" and believe that it affects everything.
The nontheist or materialist (i.e., someone who believes the world and all that is within it came about by natural processes) is not excused from explaining what went wrong. Too often, those representing that camp offer up questions without providing sufficient answers from their own worldview. For example, one question I would ask the materialist is how peace is defined in a materialistic worldview. Is it even a meaningful construct and if so, how does one arrive at a moral construct such as peace? If I am to accept materialism or Darwinian evolution as a valid worldview, then it is the strong that survive. Through whatever means necessary, the stronger will override the weaker and that often will entail the exact opposite of what is commonly thought of as "peace."
Third, we need to consider what is the solution to these problems from the perspective of the various worldviews. As I indicated above, under Islam, the solution to the world's problems is through submission--often coerced--to Allah. For the Muslim, once Shariah law is properly in place, society will function better. For some materialists, their solutions are offered through better science and understanding, but to what end? Research physicians seeking a cure for cancer and a dictator seeking to destroy a race he deems to be inferior are both operating from materialist assumptions. If you believe one of them is wrong, how do you make that determination from an objective point of view? One's personal distaste is insufficient ground for moral oughtness.
Christians believe that the world was created good and that because of sin, shalom--which not only means peace, but also includes human prospering and completeness--was fractured deeply. The only way individual, societal, and universal shalom may be restored is through a Restorer. Christians believe that restorer is ultimately found in the person of Jesus Christ. Christ came to redeem the lost and ultimately will restore heaven and earth to its original goodness. Peace will be restored.
Fourth, we must not ignore what has happened in history. A common assertion made by those who would tear down religion is that religious systems are at the root of most major world conflicts. For example, last month, President Obama equated Islamic extremism with the Christian response during the Crusades, which demonstrated a poor understanding of history. The Crusades were an attempt to push back Islamic expansion and jihad--it was first defensive, not offensive. However, it would be unfair to minimize any atrocities conducted by Christians during that time. Christians are called by Jesus to love their neighbors and when they do not act that way, it acts against their worldview rather than consistently with their worldview as might be the case with Islam or, frankly, with nontheistic materialism.
If one explores the history of thought, Christians have been at the forefront of developments in technology, science, education, human rights, and many other things. Christians are called to steward the created order in a way that promotes love for God and love for neighbor. Admittedly, there have been things done in the name of Christ that were inconsistent with the Christian worldview, but it is unfair to minimize the many positive contributions by Christians.
When considering historical facts regarding peace, we cannot ignore the 20th century. The number of people killed by those operating from an atheistic worldview far exceed the atrocities done in the name of Christ. Hitler, Stalin, Pol Pot, Margaret Sanger are just the tip of the iceberg.
In sum, when considering the statement about why religions don't achieve peace, ask yourself: 1) Does my worldview explain why there is a lack of peace and how we can achieve it?, 2) In my worldview, is peace even a meaningful construct?, 3) what does a balanced view of history tell me about promotion of peace?, and 4) what am I personally doing today to promote shalom?
For further reference, consider reading:
Why You Think the Way You Do by Glenn Sunshine
The Victory of Reason by Rodney Stark