Here, LB, the information is not that hard to find: Global Security is a non-biased source.
http://www.globalsecurity.org/military/ops/iraq_orbat_coalition.htm
Iraq Coalition Troops
Non-US Forces in Iraq - February 2007
The size and capabilities of the Coalition forces involved in operations in Iraq has been a subject of much debate, confusion, and at times exageration. As of August 23, 2006, there were 21 non-U.S. military forces contributing armed forces to the Coalition in Iraq. These 21 countries were: Albania, Armenia, Australia, Azerbaijan, Bosnia-Herzegovina, Bulgaria, Czech Republic, Denmark, El Salvador, Estonia, Georgia, Kazakhstan, Latvia, Lithuania, Macedonia, Moldova, Mongolia, Poland, Romania, South Korea, and the United Kingdom.
However, in the August 23, 2006 Iraq Weekly Status Report (Slide 27) the State Department listed 27 foreign countries as contributing troops to the Coalition in Iraq. The additional four countries were Japan, Portugal, Singapore and the Ukraine.
In addition, that same Weekly Status Report listed 34 countries (including the US) as maintaining personnel in Iraq (as part of the Coalition, UNAMI, or NATO). The State Department reported that Fiji was contributing troops though UNAMI and that Hungary, Iceland, Slovenia, and Turkey were assisting with the NATO training mission. However, it is unclear whether Hungary actually maintained any forces in Iraq as part of NATO or UNAMI since its government announced the complete withdrawal of troops in December 2004.
In testimony before the Senate Armed Services committee on August 3, 2006, Secretary of Defense Rumsfeld described the coalition in Iraq as composed of 34 allies (plus the US).
As of June 13, 2006, MNF-I reported that 27 countries (including the US) maintained responsibility over the six major areas of Iraq. Since that time, Japan has withdrawn all of its forces from Iraq.
For the purposes of this tally, only countries that contribute troops as part of Operation Iraqi Freedom are counted.
Countries which had troops in or supported operations in Iraq at one point but have pulled out since: Nicaragua (Feb. 2004); Spain (late-Apr. 2004); Dominican Republic (early-May 2004); Honduras (late-May 2004); Philippines (~Jul. 19, 2004); Thailand (late-Aug. 2004); New Zealand (late Sep. 2004); Tonga (mid-Dec. 2004) Portugal (mid-Feb. 2005); The Netherlands (Mar. 2005); Hungary (Mar. 2005); Singapore (Mar. 2005); Norway (Oct. 2005); Ukraine (Dec. 2005); Japan (July 17, 2006); Italy (Nov. 2006); Slovakia (Jan 2007).
Countries planning to withdraw from Iraq: Poland had earlier claimed that it would withdraw all soldiers by the end of 2006. It however extended the mandate of its contingent through at least mid-2007. Denmark announced that it would withdraw its troop contingent by August 2007.
Countries which have recently reduced or are planning to reduce their troop commitment: South Korea is planning to withdraw up to 1000 soldiers by the end of 2006. Poland withdrew 700 soldiers in Feb. 2005. Between May 2005 and May 2006, the United Kingdom reduced the size of its contingent by 1,300. The United Kingdom also is planning to reduce significantly the size of its contingent by the end of 2007, with an initial reduction of 1,600 troops followed by an additional 500 troops by end of 2007.
http://www.globalsecurity.org/military/ops/iraq_orbat_coalition.htm