To say the Continentals did not use bayonets at all is being a bit too broad. Two factors contributed to Continental regiments seldom making widespread use of bayonet assaults early in the war (besides the occasional rifle unit, like those /u/bodark43 described in his post, whose weapons generally lacked bayonet mounts). These are:

1) Lack of supply: The American colonies had no military industry to speak of in 1775. Most Americans carried imported muskets, those left over from the French and Indian War (both British and French), and a relatively small number of American knock-offs of the Brown Bess. While there were any number of reasons for a civilian to own a musket (militia duty, home defense, hunting/sport), there was little cause for someone to purchase a bayonet. Thus, there were few bayonets in the colonies, and only a limited capacity to produce new ones.

2. Bayonet charges take great discipline to work: Running at a group of armed men armed with a minor improvement over the pointed stick is, generally speaking, a really bad idea. An effective charge relies on men advancing at a quick but even cadence with a commitment to carry the attack home. The most effective way to achieve this, according to military thought at the time, was long hours of drill, machine-like obedience to orders, and a strong sense of espirt de corps within a regiment. In the early part of the Revolution, the Continental Army lacked these traits. Enlistments were short, the men were often fussy and quarrelsome, there was no unified drill system for the army, and few officers had the expertise or power to drill their many in any of the systems floating around the army.

In 1777-78, this began to change. The loss of New York City at its attendant military embarrassments, along with the failed attacks into Quebec, showed that the war could not be won by a temperamental force of part-timers. The push towards professionalization was sped along by the loss of Philadelphia and the army's inability to make another winter campaign out of Valley Forge.

1778 saw both the formal establishment of the French alliance and the appointment of Baron von Steuben as Inspector General of the army. France poured money and equipment into the Continental Army, including a fresh supply of bayonets. Von Steuben worked out both a practical drill manual for the army and an effective enough training method to spread it quickly.

By the Battle of Monmouth in June of 1778, Continental formations were no longer melting away at the sight of British bayonets. In 1779 (Stony Point) and 1781 (Yorktown), Continental light infantry launched successful assaults on fortified British positions using nothing but the bayonet.

For a good survey of the growth of conventional military capacity in the Continental Army, see Martin and Lender, A Respectable Army.