My first answer on this subreddit (and on reddit), so hopefully I'm doing everything correctly. I've written several papers at postgraduate level on Austrian German/Czech relations in the Austro-Hungarian Empire, so I have a reasonable understanding of this period (even though my focus was on the Austrian German side and mostly in the twentieth century part of this period).

Your family story is quite plausible and there were many Czechs that left the Empire during this period. There could be any number of reasons to explain the tensions which caused your family to leave, but I can suggest the most likely. The first possibility which jumps to mind is that the Austrian Empire fought a war with Prussia in 1866, which Prussia won decisively. Much of the war was fought in Bohemia, so it's possible that this war was a direct motivation for your family to leave. However, I'm assuming based on "tensions" that more political reasons are responsible. It probably had something to do with the Austrian Compromise of 1867, and tensions this created between Czechs and Austrian Germans in the Czech lands (the two provinces Bohemia and Moravia, which comprise the modern day Czech Republic).

I want to start by giving a little bit of background about Germans in this period. Generally people use the word "German" to refer to citizens of modern Germany, or one of its predecessor states. This can lead to a lot of confusion, because roughly 25% of Germans (ie German-speakers) in Europe during this period lived outside Imperial Germany. Similarly, "Austrian" can refer to citizens of modern Austria, but also citizens of the Austrian Empire, which would include non-German speakers like Czechs. Historians can instead talk about Reichsdeutsche (citizens of Imperial Germany), Volksdeutsche (ethnic Germans), or give another more specific description like German Bohemian (German-speakers living in Bohemia). Most German-speakers in the Austrian/Austro-Hungarian Empire identified as Austrian in terms of citizenship and also as German in the sense that they were part of a larger German-speaking nation. My preference is to use Austrian Germans to refer to German-speakers living in the Austrian/Austro-Hungarian Empire in this period.

A little context on the tensions I'll be discussing. There were a number of revolutions across Europe in 1848, most of which failed. Generally speaking, people supported the introduction of more civil liberties and the establishment of democratic governments in their countries. Large numbers of people, so-called "Forty-Eighters", emigrated after becoming disillusioned by these failures. The United States was the largest recipient of these migrants. I see this wave of emigration, which included many Czechs, as a precursor to the emigration which came later in the period in which you’re interested.

So what happened in the Czech lands after the failed 1848 revolutions? The new Austrian Emperor, Franz Joseph (the same one who was ruling during the First World War), returned Austria to absolutist, centralised rule in the 1850s, in which Austrian Germans dominated political life. Czechs were only eligible for a third of the seats in the local Bohemian and Moravian parliaments, despite comprising a majority of the population in both. German was also the de facto official language in the Czech lands, although it was only spoken by roughly one-third of the population. That being said, it’s worth noting that many people were bilingual, and this was quite normal in the Empire at the time.

Most Czechs supported remaining part of the Austrian Empire, but instead favoured reforms which would mean greater autonomy and self-rule for the Czech lands within the Empire. A number of Czech politicians argued that Czechs should have their education conducted in the Czech language, and that state administration should deal with each person in their language of choice. Basically they wanted the Czech lands to be officially bilingual rather than dominated by German-speakers. These demands were made for both political and economic reasons. Obviously there was an element of national pride, but there would be economic benefits too, as more employment opportunities opened to Czechs who could not speak German.

Like the Czechs, other ethnic groups in the Empire also sought greater federalisation and autonomy. However, it was the Hungarians, as the second largest ethnic group in the Empire after the Austrian Germans, who made the most compelling argument for reforms. After the Austro-Prussian war of 1866, the Hungarian nobility exploited the Empire's weakness to demand more power within the Empire. The Emperor felt that the only way to restore Austria as a great power was to share his power with the Hungarian nobility, and thereby control the other ethnic groups in the Empire. The result of this was the so-called Austro-Hungarian compromise (Ausgleich) which was signed in 1867. The Ausgleich split the Empire in half, one half dominated by Austrian Germans and the other half by Hungarians. These functioned as two equal Kingdoms joined as one Empire. This is why many historians refer to Austria-Hungary as the "Dual monarchy". The Czech lands ended up in the German part of the Empire (also known as Cisleithania).

As the third largest ethnic group in the Empire, many Czechs had also advocated for more power and equality within the Empire. The leading Czech politician of the era, František Palacký, wrote a book in 1865 warning against the adoption of Dualism, because he thought it would only encourage linguistic nationalism among the other ethnic groups, and this would lead inevitably to the collapse of the Empire. Czech politicians had advocated instead for 'Trialism'; that being, self-rule for the Czech lands within the Empire, and equal status for Czechs with Austrian Germans and Hungarians. As it was, Czechs had representation in the provincial assemblies in Bohemia and Moravia, and the parliament in Vienna, but these were all still dominated by the Austrian Germans. Czechs also wanted equal language rights in the Czech lands. The immediate result of the Ausgleich was that, when the Austrian parliament reopened in 1868, all the Czech representatives walked out in protest. Some returned to parliament in the early 1870s, but most Czech representatives continued to boycott the Austrian parliament until 1879. Obviously they felt that their complaints about inequality and injustice were significant enough to boycott parliament for 10 years.

Most of my answer so far has been about politicians, but how did this impact daily life for ordinary people in the Czech lands? For most of the nineteenth century it was quite normal for Austrian Germans and Czechs to live together bilingually in the Czech lands, but the Ausgleich led to increasing radicalisation on both sides. Ironically this was caused largely by the well-meaning but misguided government attempts to reduce tensions. There were a number of attempts at reform from the 1870s, none of which had much success at resolving the disagreements. The government started using the national census to categorise people by language either as German or Czech, and used this data to determine things like funding for German or Czech language schools in particular areas. Previously many people identified either regionally, as Bohemian, or by citizenship, as Austrian, but increasingly nationalism was determined solely on language grounds. This decision forced bilingual and apathetic people to choose sides in the ongoing conflict about national identiy, which had previously been quite insignificant. Nationalists on both sides were quick to exploit things. For example, German-language activists started fundraising campaigns to pay Austrian German families to move and settle in Czech-speaking towns, and thereby force the government to fund German language schools there. These actions were viewed with much hostility by other residents, and such actions often led to retaliations. There are many reports from this period in which German- and Czech-speakers fought in the streets over the opening of schools, and tried to prevent people who spoke a different language from moving into their area. Over time, majority-language populations forced minority-language populations to leave, and many areas became more monolingual. Unhelpfully there was also an economic crisis in Europe during much of the 1870s, known as the Long Depression, and this likely contributed to large numbers of people emigrating from the Czech lands in this period, including your family.

Hope this was helpful. I'm happy to answer more questions about of the above, if I'm able.

Sources and further reading: Jeremy King's 'Budweisers into Czechs and Germans' is a great read on how national identities were formed in Bohemia during this period. I also highly recommend Pieter Judson's 'Guardians of the Nation: Activists on the Language Frontiers of Imperial Austria', which discusses how national identities were used for political purposes in the Empire during this period. Both of these books are very accessible and readable. For more in-depth reading I can recommend Pieter Judson's 'Exclusive Revolutionaries: Liberal Politics, Social Experience, and National Identity in the Austrian Empire, 1848-1914'.