There is no room in this country for hyphenated Americanism.
Interestingly, he was not talking about what people now claim he was talking about. It was spoken during WWI, when there was a strong suspicion that German-Americans and some Irish-Americans had close political ties with the nations of their birth, rather than America.
Do you suppose for one moment that a person who calls themselves "Irish-American" today plots with the government of Ireland against the USA, or that a "German-American" is plotting the next German uprising? Are "African-Americans" against America and have some political or national interests in an invasion of America by "Africa?" This was the concern that Roosevelt was speaking of. Not people who today use the term to simply denote their heritage or to take pride in it. Unless you suspect that I, being a "Welsh-American," am secretly indebted to Wales and seek America's destruction by Wales, I am not the person Roosevelt was talking about.
By the by, the time period was one of intense 'Nativism' and Roosevelt was speaking to Catholic placatingly, as they had recently been made subject of various violent attempts to exclude them from citizenship, politics, and land-ownership, which is where the Knights of Columbus came from - a response to armed Protestant men's groups who literally attacked Catholics on sight and burned down Catholic churches; in America and in the name of Nativism. These same people who believed that an Irish-American was no American at all also believe that a Catholic-American took his or her marching orders from the Pope and hope to destroy our nation.
Do you really want to align yourself with those people? That was the group that Roosevelt was attempting to explain were not so bad at all.