The sharp decline in Iraq violence in the last two years has not eliminated the cause of the killing spree, or the many people who want to revive it. It comes down to this. For several centuries, the Sunni Arab minority (about 20 percent of the population, until recently) ran the country. They are still there, and still believe they can regain control. The Shia Arab majority (over 60 percent of the population) are willing to kill all of the Sunni Arabs to ensure that this does not happen. All this goes back a long way.
Iraq is an ancient civilization that has been subjected to near constant foreign occupation (Mongol, Iranian, Turkish, British) for the last thousand years. What we know as Iraq was put together by the British, in the 1920s, from fragments of the recently dissolved Turkish Ottoman empire. The northern part of Iraq, containing mainly Kurds, was then considered part of Turkey itself, and not an imperial province like the rest of Iraq. But there was oil up there, and the British did not want the Turks to have that, in case there was an effort to revive the Ottoman empire.
The British set up a constitutional monarchy in Iraq, complete with parliament, and royal family imported from Saudi Arabia (a noble family that had been ousted by the Sauds). While democracy was alien to this part of the world, many Iraqis took to it. But there were serious problems with corruption, plus the tribal, ethnic and religious rivalries. The Kurds weren't Arab (they were Indo-European, and about 20 percent of the population), 60 percent of the Moslems were Shia (a sect considered heretical by the conservative mainline Sunnis). The Sunni Arabs may have been a minority, but they dominated commerce, government, education and running things in general. Since the 16th century (when the Iranians were driven out), the Sunni Turks had relied on the Baghdadi Sunni Arabs to help administer the area.
Britain had to re-occupy Iraq during World War II, because the Sunni Arab dominated government (not the king) tried to ally itself with the Nazis. At the time, many Arabs admired Nazism. Many still do. The Brits again conquered the country, using three divisions and taking three weeks to do it. Britain found another bunch of Sunni Arab notables and told them they could run things if they stayed away from the Nazis. That lasted for about a decade, until the Sunni Arab politicians and generals decided that this democracy stuff wasn't working for them. The royal family was massacred and parliament purged of "disloyal" elements. The Sunni Arabs were back in absolute charge, via a series of dictators, until Saddam Hussein was deposed in 2003.
Saddam was a particularly brutal tyrant, killing hundreds of thousands of Iraqis (mostly Shia and Kurds), and terrorizing nearly everyone. After being run out of Kuwait in 1991, and barely surviving another Shia rebellion, he made peace with the Sunni Arab tribal leaders, and unleashed yet another terror campaign on the Shia Arabs. The Kurds were now independent, protected by British and American warplanes. So Saddam accepted that, and left them alone.
Now, this is the critical thing that many Americans don't understand, or even know. When Saddam was deposed in 2003, most (well, many) Sunni Arabs believed they would only be out of power temporarily. This sort of thing you can pick up on the Internet (OK, mostly on Arab language message boards, but it's out there). Saddam's followers (the Baath Party) and al Qaeda believed a few years of terror would subdue the Shia, scare away the Americans, and the Sunni Arabs would return to their natural state as the rulers of Iraq. U.S. troops quickly picked up on this Sunni mindset. Because Sunni Arabs were the best educated group, most of the local translators the troops used were Sunni Arabs, and even these guys took it for granted that, eventually, the Sunni Arabs would have to be in charge if the country were to function. The Sunni Arabs believed the Shia were a bunch of ignorant, excitable, inept (and so on) scum who could never run a government. Four years later, the Shia had sort-of proved the Sunni Arabs wrong. At that point, many Sunni Arabs wanted to make peace, not suicide bombs.
Which brings up another major issue in Iraq. Many Iraqis believe only a dictator can run the country, and force all the factions to behave. However, a majority of Iraqis recognize that dictatorships tend to be poor and repressive, while democracies are prosperous and pleasant. The problem is that the traditions of tribalism and corruption (everything, and everyone, has their price) do not mesh well with democracy. This doesn't mean democracy can't work under these conditions, many do. It does mean that it takes more effort, and the results are not neat and clean, as Americans expect their democracies to be.
Until the Iraqis develop "civil society" (rule of law and economic freedom), they will be at risk of increased violence. There are still thousands of Sunni Arabs willing, and able, to resume the terrorism. All they need is the support of many, not even most, Sunni Arabs. But right now, the Sunni Arab population is terrorized. When the Sunni Arab terrorists (and al Qaeda supplied foreign volunteers) instigated their terror campaign against the Shia Arabs, the Shia struck back in kind. To the surprise of the Sunni Arabs, the Shia used their control of the security forces, to slaughter tens of thousands of Sunni Arabs. This led to about half of the Sunni Arabs fleeing their homes, and half of those fled the country. By 2007, the Sunni Arabs were only fifteen percent of the population.
Many Sunni Arabs are now more concerned with survival, than taking back control of the country. The Shia led government has suppressed the Shia death squads, but the Shia killers are still there. And the largely Shia security forces are not gentle with Sunni Arabs. There is also the threat of Shia extremism, backed by the Shia radicals that run Iran. This sort of thing has little support within Iraq, but the Shia radicals are, well, radical, and not always rational in their decision making. An Iraqi government dominated by radical Shia would be bad news for the Sunni Arabs (and not too good for the Kurds, who are also largely Sunni.)
The ethnic and religious violence in Iraq has stopped (not completely), not gone away.