Have there been any efforts by the countries of Europe to restore their monarchies?

Bavaria stopped being a monarchy not even 100 years ago (1918). After that there were attempts to reinstate the monarchy (Monarchism in Bavaria after 1918). Sadly most of these attempts were destroyed by the Nazis (which gives me my very own reason to hate them). If any of these attempts had succeed, maybe WWII could have been delayed and the severity reduced. But it's likely that Adi would have just turned Bavaria into a wasteland.

I'm usually not a monarchist, I think that monarchy is only good if the king is great (which is often not the case). Most of the rulers of Bavaria throughout history were capable leaders who loved their country, not their power. Even today, the official heir of Bavaria would be a great king, which in my opinion is shown even more so because he doesn't want to be king.

In Romania, as recent events has shown (during the funerals of HM King Michael I) there is a strong monarchist sentiment among the "profound Romania", the deep, usually silent part of the Romanian populace.

The vast majority of Romanians claim that the Revolution of 1989 is an "unfinished revolution", one that only overthrow Ceaușescu, instead of finishing Communism for good. The obscure and corrupt combination of former politcal police officers and high Communist Party members, now disguised as "investors" and "democratic politicians", is seen as the main reason of the country's mainly negative performance in the last 30 years.

Conversely, the Monarchy (through the King's elder daughter, Princess Margareta) is regarded as the main guarantee for the country's democratic future, stability and conservation of true nation values. So YES, Romanian Monarchy's restoration is possible and very desirable for the political stability of Eastern Europe.

The main reasons why it hasn't happened yet are:

1. an insufficiently informed populace, often a victim of former (42 years of tough Communism...) and present Leftist propaganda, which is still very widespread ("fake news", often gross lies, even in mainstream media)
2. former political police officers and other kind of nomenklature still controlling all political parties, some NGOs, media outlets and the vast part of economic resources
3. insufficient understanding of the "Romanian Royal issue" and support for it from the West.

Big part of the reason of the existence of sovereign states in Eastern Europe is nationalism. However, in many cases these countries did not have own national ruling families. For example Austrian Habsburgs ruled Hungary, Bohemia, Croatia and the territories now known as Slovenia and Slovakia, so most of these countries did not want a foreign monarch back. Poland (and Lithuania) was an elective monarchy before its partition, so there was no real royal house. I don't know if Estonia or Latvia had a royal house at all. Same with Belorussia.

There are a bunch of monarchists in modern Russia. Some are "theoretical" monarchists, some want to bring back Romanovs, some want Putin to be the Tzar.

• Monarchist party in Russia launched

• Wikipedia List of monarchist organizations - Russia

• An article on monarchists in Russia:

  At various times, politicians from across the political spectrum have endorsed constitutional monarchy for Russia, including the former Union of Rightist Forces co-Chairman Boris Nemtsov, Liberal Democratic Party of Russia head Vladimir Zhirinovsky, and St. Petersburg Governor Valentina Matvienko [ DVK note: For those who don't know Russian politics, these are fairly high level figures in russian politics ]

  ... Many intellectuals and cultural icons have also jumped on the monarchy bandwagon. Two of Russia's most popular filmmakers, Nikita Mikhalkov and Stanislav Govorukhin, have paraded their monarchist colors. Stanislav Belkovsky, the founder of the National Strategy Institute, said in February 2005: "I believe that the restoration of the monarchy, either formally or informally, is the only choice for Russia, since it is the only way to restore the sanctity of the supreme power."

  ... A September poll by the All-Russia Center for the Study of Public Opinion (VTsIOM) indicated that 19 percent of Russians agreed with restoring the monarchy, but only if an acceptable candidate can be found. Support is higher in Moscow and St. Petersburg.

The article contains some further info and analysis as well.

Are you restricting your question to a certain time period? If not, then there are a number of examples, besides those already given in the other answers, which could qualify:

• France has vacillated between a republic and a monarchy several times; the First Republic was succeeded by the First Empire under Napoleon, after which the Bourbon kingdom was restored. The Second Republic was succeeded by a restoration of the Empire.
• In the 17th century, the three monarchies of England, Scotland, and Ireland (in a personal union) were deposed, and replaced with a unitary republic from 1653 to 1659. The three monarchies were restored in 1660.
• Spain has also wavered between a republic and a monarchy. The First Republic was established in 1873, but the Bourbon monarchy was restored the following year. The Second Republic existed from 1931 to 1939, after which the Bourbon monarchy was again (eventually) restored.

There was a referendum to restore the Albanian monarchy in 1997, following a period of severe unrest. The official results of the referendum had the motion failing by a 2-1 margin; the monarchists then claimed that the result was invalid, riots broke out, and the crown prince Leka fled the country and was tried in absentia for inciting rebellion.

The country subsequently adopted a new republican constitution in 1998. Leka was later pardoned and returned to the country, but monarchist ardor seemed to have cooled somewhat:

Fewer than 500 people turned out to greet him. His supporters had been hoping he would be greeted by a crowd of several thousand.

Leka died in 2011, and it does not appear that Leka II (his son and the current pretender) is currently pressing the monarchist cause:

After the election of Bujar Nishani as President in 2012, Leka [II] was appointed as political adviser to the President.

Simeon Saxe-Coburg-Gotha was Tsar of Bulgaria from 1943 to 1946. 2001, Simeon resumed the role of leader of the nation upon taking office as Prime Minister of the Republic of Bulgaria from July 2001 until August 2005.

But there were no plans of a restoration of the Bulgarian monarchy.