Actually, the major complaint of the colonists was that they were denied their "rights as Englishmen" as guaranteed under Magna Carta and the English Bill of Rights of 1688. They DID NOT want representation in Parliament, as the above post implies; such representation would be infeasible considering the distances involved. What they did want--and demand--was that any taxes levied on them be levied by their duly elected representatives, in this case the colonial legislatures. They had paid taxes levied by the legislatures in the past; and had not raised major objections to the Navigation Acts which had been instituted for regulatory purposes only. They did resent--bitterly--Parliament's attempts to tax them for revenue purposes only. This they believed was the sole province of their assemblies.
Also, they believed their rights to trial by juries of their peers was violated when customs cases were transferred to England. Colonial juries uniformly found in favor of defendants charged with customs violations, and the British transferred jurisdiction to British courts to increase the chances of conviction. This to the colonists violated their rights under Magna Carta.
In the Declaration of Independence, Jefferson lists a number of offenses by George III in great detail. The true sense of violation of their rights--as Englishmen as well as their natural rights--is as set forth above. It is well worth noting that even after the war commenced, the colonists sent George III an Olive Branch Petition offering to return to his governance if they were only granted their rights as Englishmen.
As we can see in the Declaration of Independence, Americans believed they had certain inalienable natural rights. They believed that they had the right to be governed only if they consented to be governed. They believed that any government that ruled them had to protect their right to life, liberty, and property. The colonists believed that the British were infringing on three of these four rights.
First, they believed the British were failing to secure their consent to be governed. The British did not let them have representation in Parliament. This, to the Americans, was a failure to obtain their consent.
Second, they believed that the British infringed on their liberty. They felt that British practices like the use of vice-admiralty courts constituted an infringment on people's liberty. Finally, they believed the British were violating the sanctity of their property. This could be seen in such British actions as quartering soldiers in private homes or in the issuance of writs of assistance.
In these ways, the American colonists felt that the British were violating their natural rights.