Were there any well known royal dynasties that did not in some way cite religious mandate for their rule?

All of them were blessed by the clergy, (if not, they would have too little power to be kings). So, in some way all had that mandate, because it was foolish not to get it. Maybe, somebody refused. So, we need an actively atheistic king...even in Napoleon's empire clergy blessed the power of the Emperor. I think, not only the king, but the rulers, who hadn't sacred power were early Bolshevik's chiefs - Sverdlov and Lenin. Because, they were sacred in their, communistic religion, they didn't need any blessing from the opponents.

An official king, that was not officially sacred (but had his blessings) was Jiri z Podebrad - the czech king after Husits' Wars. He was simply elected. ruled for 13 years. His son wasn't a king.

Most lesser rulers in the German Empire were not kings, but suit your demands. "Simple" counts, princes and so on. They were not sacred, could not heal by touch and so on. Their power over people was absolute, but not over clergy - it was too powerful there and then. And their countries were often larger than Manhattan or even Rod Island and a dozen had more than a million people.

There could be different notions of "god-santioned power" so I will list them.

1. The king is thought to originate from gods by his ancestral line or claimed that the gods directly interacted with the originator of the dynasty and gave them the right to rule. This is the most strong idea of the god-santioned power.

2. The gods consented to the king's rule. In many ancient republics and monarchies there was a rite after the king's election by the people and/or choosing the heir in a monarchy, which was intended to test whether the gods accepted and loyal to the pretender. It was usually a chance-based experiment through which the gods should express their will. This kind of god-santioned power meant that the gods were loyal only to the current ruler and not to the dynasty as a whole.

3. There is a usual religious rite of becoming a king, like a marriage for example. The ceremony does not mean the gods somehow support exactly this ruler, but rather intended to make the ruler to take responsibility before the face of the god(s). If he becomes a bad ruler, he may face punishment in afterlife. The ceremony usually takes form of an oath, promise, possibly involving a religious scripture. This form of rite is used in many modern republics as well, such as the United States for example, where the president makes an oath on the Bible.

I think the most ancient form was the form 2 which still can be found in savage tribes. This possibly came after some disasters or contested power which indicated that the gods were not always positive regarding the ruler.

The form 1 evidently came with rise of large national states where the idea of a king as a family leader did not work any more. The people (sometimes consisting of conquered peoples) heeded an explanation why this dynasty is better for all, not just for their relatives.

The third form originated from form 1 when the dynastic lineage was interrupted or with the order changed to the republic. The king was not able any longer base his power on a divine origin.

Sometimes the form 1 was skipped and form 2 directly converted to form 3 either when a capable ruler went into conflict with clergy who previously could manipulate the outcomes or when the society became multi-religious so that an oath before the own gods of the ruler could convince even those who belonged to a different religion that he will keep the promise.

Muslim monarchies generally don’t rely on any divine mandate or right, just pragmatic concerns (i.e. obeying the ruler is the best way to preserve peace and ensure God’s laws are implemented).

The ruler is expected to apply the divine law (Shari’ah) but this is a duty on the ruler and a condition for his legitimacy rather than vice versa. Generally, Muslim scholars/ideologists justified the right to rule on the basis of a ruler’s ability to command loyalty from the centers of power, e.g. a powerful clan (this loyalty base was called shawka which roughly means strength or backbone). Naturally, nothing proves shawka better than actually winning power, and if your shawka is based around a clan than that will tie in nicely with hereditary rule.

It’s then a question of Shari’ah law as to when it becomes permissible to revolt against a ruler who fails in his obligations but the consensus has always been that obedience is required except in the most extreme cases. For example the scholars of Islamic Iberia (Al-Andalus) supported deposing their local monarchs in favor of the Almoravids of Morocco because they believed the latter were the only hope for protecting the Muslims from the Christian kingdoms. In modern times, King Saud of Saudi Arabia was deposed by the royal family in favor of his brother Faisal. The religious scholars supported this decision and issued their own edict, which cited nothing more than the consensus of the royal family and the religious scholars and the need to avoid strife.

There were attempts to come up with something like a divine mandate, e.g. the jabriyya movement in the Umayyad era, which said that objecting to Umayyad rule was tantamount to objecting to God’s will since he caused them to be in power, but this did not catch on. It was also said that the Mongol conqueror of Baghdad was warned that executing the last Abbasid caliph would bring forth calamities and divine retribution but other (Shi’ite) scholars reassured him that this had no basis. It was more of a folk belief than an official ideology.

Finally, I should mention that Twelver Shi’ism believes the twelve Imams to be divinely appointed, but this is purely theoretical; only the first two held political power (and the second one promptly abdicated “for the greater good”) and the doctrine was only formulated after their time.

See Patricia Crone, Medieval Islamic Political Thought and Hugh Kennedy, Caliphate: History of an Idea

There was the rulers of the Gupta Empire. As I said in a previous answer, the empire existed from 3rd century A.D. and 550 A.D & the main dominant 'religion' of the empire was a nontheistic branch of Hinduism called classical Mimamsa that believes the gods only exist as ideas, not real beings, and doing rituals/social duties are how one lives a better life and represent dharma, so you literally couldn't claim to be a monarch because 'the gods wanted me on the throne'. The empire was about 660,000 square miles in size around 440 AD. Allahabad Pillar inscription mention that rulers of several frontier kingdoms all paid tribute the Gupta monarchs. These rulers, in turn, provided education, built roads, and issued money in the form of gold coins. The monarchs also remained leaders due to their military innovations which helped to protect the empire from enemies like the Huna peoples. So basically, being good logistic leaders, warriors, and scholars (as well as tolerance of minorities who didn't believe in their nontheistic Hindu philosophy) legitimized their rule.

Since the gods were thought to be creators of the world, it was quite difficult for ancient people to think the gods were unrelated to the power in some manner. Any leader (not only king but also a general) had to convince their people and soldiers that the gods were at least loyal to their side to maintain good morale. Also in the ancient states often the only two branches of power were the religious one and executive one, thus the inauguration was performed by the supreme priest as the second-important figure in the state.

That said, the Roman emperors did not claim the divine mandate. Their power was in theory delegated by the people and senate of Rome. You of course heard that they were proclaimed "divine" sometimes, but usually post-mortem, and this was a honor conferred by the decision of the senate (and did not imply the dead were gods but just god-like). There were also temples of the "Emperor's genius", genius being a minor god, personal protector of the emperor (all people were thought to possess a genius as well). That is the people just honored the personal protector of the emperor so that his life to be safe.

Some emperors just like other noble Romans traced their ancestry to gods, but this was never used to justify their special rights to the power.

But I am not sure that this case falls under your request because the empire was officially a republic until the reign of Heraclius who after he defeated the Sassanid Empire adopted the title "King of Kings" (Basileos Basileon) which was previously held by Khosrov II, the defeated Sassanid king. It should be noted meanwhile that adopting a foreign royal title was not that charged in Roman Empire/Republic where sometimes even republican officials could receive a royal title from local barbarian tribes as a sign of loyalty. Thus it could be argued that even after Heraclius the power was in theory delegated to the emperor by the three forces combined: the people, the army and the church (the first emperor whose coronation involved the Patriarch though was that of Leo I the Thracian, before Heraclius).

The eastern title "King of Kings" was not religious in nature though. It just meant the king was recognized as the supreme king by other peer kings.

If we look deeper in the history, the ancient Greek kings were just gens elders. Take for example, Sparte where initially was four and later two tribes with their respective "kings", the two considered the equal kings of Sparta. I think the most ancient concept of "king" was exactly that of a tribe's elder, the head of a family.