The primary issue I think you'd run into is not knowing the riverbed. Certain portions of rivers are known to have shifting sandbars, sunken trees or debris from a shipwreck, and so on which would not be visible from the surface of the river but could easily damage a ship or you could run aground and become stuck. You would need the help of a rivermaster or river pilot to successfully navigate a river, or at the very least you would need to station people at the bow of the craft to drop lines to measure the depth of the water.
This latter method was quite common on the Mississippi in areas unknown to the pilot or where the sands were known to shift rapidly. Samuel Clemens' pen name of Mark Twain is a measurement of river depth meaning two fathoms (a generally safe depth).
If you reach a particularly sharp bend in the river, particularly swift current, or a ford, you may need to portage the ship (that is, tow it overland to where you could safely sail from) to get past it.
Mxyzplk is right about the need for alternative power, too, as depending on the wind may not be possible. Historically, again, we can see that galleys with square sails and decks of rowers could navigate rivers, but those ships date from periods when sailing was limited to coastal waters.
In an emergency or during wartime it could be done (see the Mississippi campaign and attack on New Orleans in the American Civil War and War of 1812, although also consider that strategic rivers often have chains which can prevent transit) but it is much less safe than procuring a ship designed for river travel.