The motivations for each of the kings would have been significantly different, but piety certainly played a role in all of their decisions. There would not have been much, if any, material gain to be had by going on Crusade. Crusades were extremely expensive and none of the kings would be planning to move to or take control of the Holy Land (it already had a king for one thing). There was an element of reputation/fame to be gained by going on Crusade. While the disastrous Second Crusade had not done a lot for the reputations of the kings on it the First Crusade had made legends out of its participants. A successful crusade was a great way to ensure you were remembered and respected for centuries. Sometimes even unsuccessful crusades did this (St. Louis was on two disastrous crusades) but that was less clear at the start of the Third Crusade.

The most important factor in motivating people to go on the Third Crusade was the reason it was called: Jerusalem had fallen. The loss of Jerusalem was a big deal, it's Christianity's holy city and Muslim control could have severely limited Christian pilgrimages to it. That's to say nothing of the embarrassing implications of losing the city to 'heathens' in a time when victory in battle was thought to be decided by God.

There was a huge political pressure to ensure that the Holy Land was reclaimed for Christianity and medieval kings were not immune to this. None of the kings were in so strong a political position that they could ignore this kind of pressure. Richard I ruled over a huge amount of territory, but his grip was often tenuous, he was later killed besieging a castle in his own land. Philip Augustus was more powerful than any French king had been before him but Richard I ruled more of France than he did, so he wasn't exactly on the best footing. Frederick I was ruling an Empire that had been wracked by Civil War for almost a century as a result of the Investiture Conflict. He had been elected to his throne as a compromise between two warring families and while he was quite late in his reign at this stage he wanted to guarantee that the stability he had spent his life working on was maintained. Frederick was also often at odds with the Papcy with regard to his desire to control Northern Italy so Crusade was also a decent way to appease the Pope.

There was also an element of fear. While Saladin now enjoys a very popular reputation with the West, and has done so since the later Middle Ages, a the time he was seen as almost demonic. There was a legitimate worry that he might set his sights on the West and the conquest of Rome. While in hindsight this was obviously not going to happen, and it's not clear how widely held this belief was, it certainly helped to motivate recruitment for a Western army to go and defeat him.

It's very difficult to know the exact motivations for each king but we can take make some guesses based on what happened later. Frederick is the hardest to understand because he died en route to the Crusade so we don't know how he would have behaved once he got there. While it can be tempting to see the fact that most of his army returned to the Empire after his death as indicating a lack of piety from the nobles who followed him that's not necessarily fair. The Empire was still only barely stable and the election of a new Emperor was always a contentious time. Many of them would also likely have had a role to play in picking the new Emperor and would have wanted to limit the amount of time they spent without any ruler.

Philip Augustus appears to have had the least enthusiasm for Crusading. He spent very little time in the Holy Land before returning to France. His signing of a peace treaty to go on Crusade was more a reflection of Papal policy than his enthusiasm for the movement. The Papacy frequently tried to end wars between Christian kings by getting them to both go on Crusade. They were still doing this up through the Hundred Years War, but this is one of the only times it actually worked. History's of the Third Crusade often barely mention that Philip was even there his participation was so minimal.

Richard I is known as the crusading king for a reason, he seemed to be genuinely invested in the movement. Piety was definitely an important part of this but the reputation and fame that came with his Crusade probably did not hurt.

Hopefully that answers your question. Jonathan Riley-Smith is generally the best Crusading historian for motivations for crusaders (although he does tend to err on the side of piety a bit) and his Crusades: A Short History is a good read. Thomas Asbridges The Crusades has a lot more detail on the Third Crusade in general, though. Anne-Marie Edde's Saladin is the best biography of the sultan and includes a discussion of both his reputation and the reaction to his taking Jerusalem. It's not the most approachable book if you don't already have at least a basic familiarity with the events of the Crusades, though.