Looking at the list of separatist movements it seems that all of them are focused around a region which has its own language/dialect and an associated culture. But are there any active secessionist movements where the region in question does not have a language distinct from the main country?
Apparently, most of the examples are found in... surprise... the United States.
California (NOT based on the Spanish language, so it counts)
New England Independence Movement
As a commenter noted, if you add history, you have the secession that prompted the US Civil War (and on the same page, you can add the Russian civil war, which included secessionist efforts in some areas that were Russian speaking).
(There're obviously some with separate languages - mostly Native American groups like Lacota or Hispanic-origin groups.)
Australia also has successionist movements, with the most prominent being that for Western Australia. A major stated grievance is not getting a fair deal of government resources considering what the state contributes to the country. All states in Australia primarily speak English. Non English speakers are either immigrants or indigenous people, rather than belonging to a specific area of Australia.
There's the English independence movement, which I think meets the criterion: although there are many dialects and even at least one language which are distinctive to subregions of England, the prestige dialect of the UK is one of them.
Some regions of Spain may be candidates, although it's hard to tell purely from a quick web search how active e.g. the secessionist (vs nationalist federalist) side of Canarian nationalism is currently.
Hong Kong seems like a reasonable candidate, although the question of language is complicated. Do you consider the fact that English is one of the official languages to disqualify it? If not, I think arguing for disqualification on the basis of Cantonese is unsound because Hong Kongers are a minority of Cantonese speakers.
I've saved my most controversial candidate for last. ISIS/ISIL/Daesh could be considered to be a separatist movement in that its goal is specifically to create a new state rather than to replace the governments of existing states. It's not defined by any particular dialect of Arabic.
There is an influent secessionnist movement in northern Italy, on a territory they name "Padania" (from the Po river). Main arguments for secession are economical (the south of Italy is less rich) and cultural, but Padania has no specific language and Italian is pretty much the same in Milan or in Rome.
The party Lega Nord (far-right, secessionnist for long through it seems to have evolved) is one of the main political local forces and has even joined Italian government in several Berlusconi-driven coalitions.
Transdnistria and Moldova. They use the same language but differen alphabet on principle (until recently the alphabet was the same). Also Moldova does not desire to join Romania.
ISIS and Shia Iraq. The same language and ethnicity but different religion.
Tajikistan is not in desire to join Iran despite the same language (but different script), again different religion (Shia vs Sunni).
Bosnia-Croatia-Serbia thing (the same, Serbo-Croatian language but different religions, Orthodox Christian, Catholic Christian, Muslim)
North and South Korea
Austria and Germany (I heard they somewhat dislike each other). Again, mostly, different religion (Austria is Catholic, Germany is mostly Protestant with exceptions).
Thailand, Philippines, Nigeria, Kenya - Muslim separatists speaking the same language and of the same ethnicity.
Sudan - Christian separatists against Muslims.
Usially if some regions adverse to each other, they intentionally invent differences in their languages, introduce new words, new spelling etc.